CARNES, Circuit Judge:
Just over a quarter of a century ago, in 1975, James Willie Brown and Brenda Watson went on a date to the Mark Inn Lounge in Stone Mountain, Georgia. They ate a steak and potato dinner and spent several hours drinking and dancing. Brown v. State, 250 Ga. 66, 66, 295 S.E.2d 727, 729 (1982). Afterwards Brown took Watson to an old logging road in a heavily wooded area, tied her up with a nylon cord, raped and orally sodomized her, and suffocated her by forcing her panties so far down her throat that they were not discovered until the autopsy. Id. She was the third woman Brown had attacked, but the other two were fortunate enough to have escaped with their lives. Id. at 73, 295 S.E.2d at 734. Because Watson did not, Brown was charged with capital murder.
The trial of the case was delayed for six years because of concerns about Brown’s mental competency. When the case did go to trial in 1981, a jury convicted Brown and sentenced him to death. That conviction and death sentence were affirmed by the Georgia Supreme Court, id., state collateral relief was denied, and the United States Supreme Court denied certiorari review. Brown v. Dodd, 484 U.S. 874, 108 S.Ct. 33, 98 L.Ed.2d 164 (1987). Brown was more successful in seeking federal ha-beas relief. In 1988 the United States District Court for the Northern District of Georgia issued the writ, effectively requiring a new trial, on Pate v. Robinson, 383 U.S. 375, 86 S.Ct. 836, 15 L.Ed.2d 815 (1966), and related Brady v. Maryland, 373 U.S. 83, 83 S.Ct. 1194, 10 L.Ed.2d 215 (1963), grounds. After conducting proceedings to ensure that Brown was competent, the State retried him in 1990. The evidence at the retrial, like that at the initial trial, was overwhelming, and the jury convicted Brown and sentenced him to death again. The Georgia Supreme Court affirmed that new conviction and sentence. Brown v. State, 261 Ga. 66, 401 S.E.2d 492, cert. denied, 502 U.S. 906, 112 [1311]*1311S.Ct. 296, 116 L.Ed.2d 240 (1991). State collateral relief was denied as well. After the Supreme Court denied certiorari, Brown v. Turpin, 519 U.S. 1098, 117 S.Ct. 781, 136 L.Ed.2d 725 (1997), Brown sought federal habeas review which was denied in November 2000. This is the appeal from that denial. Brown raises a number of issues, several of which deserve discussion.1
I. THE INEFFECTIVE ASSISTANCE CLAIMS
Brown raises ineffective assistance claims relating to two witnesses who testified at the sentence stage, Carl “White and Anita Tucker. We hold that even assuming that Brown’s trial counsel should have discovered and used certain impeachment evidence to attack the testimony of these witnesses, the state habeas court’s conclusion that counsel’s failure to do so did not prejudice Brown’s sentence is objectively reasonable.
A. Witness White
During the sentence phase, "White testified that he was the officer who booked Brown on the murder charge on May 15, 1975, and that he did not notice anything unusual about Brown’s demeanor or ability to communicate. White also testified about something Brown said: “"When I got through taking the information, as I recall, he made a statement that he wasn’t worried about this charge, that he would plead insanity and be out in a few years.”
Brown contends that his trial counsel should have discovered and used other evidence to impeach the testimony of "White. The other evidence consists of documents in the State’s files indicating that White did not report Brown’s statement until six years after it was made, and that he did so then only after an investigator, at the prosecutor’s direction, questioned White about whether Brown had made any statements while being booked that might shed some light on his mental state at that time. Brown argues that delay in reporting the statement could have been used to impeach "White’s testimony that Brown made the statement, because if such a statement really had been made, White would not have kept quiet about it for six years.
Since White had testified at Brown’s competency hearing about the statement made at booking, Brown argues that his trial counsel should have anticipated that testimony would be repeated at trial and set about to find a way to impeach it. If counsel had looked, Brown says, he could have discovered the documents indicating "White’s six-year silence about the statement. The failure to discover and use that evidence to impeach White’s testimony at trial was ineffective assistance of counsel, Brown argues. He asserts this ineffective assistance claim as to sentencing, not as to the guilt stage.
The state habeas court concluded that trial counsel performed deficiently in not discovering and using the fact that White had not reported to the prosecutor Brown’s statement at booking soon after it was made. The court denied relief, however, on the ground that no reasonable [1312]*1312probability existed that Brown’s sentence would have been different had counsel discovered and used the fact of the delay in reporting to attack White’s testimony.
In this federal habeas proceeding, the district court agreed with the state habeas court that Brown had not established prejudice from his trial counsel’s failure to discover the information and use it to impeach White. Like the state court, the district court reasoned that even apart from White’s testimony, the other evidence supporting the death sentence was overwhelming. It followed that the state habe-as court’s decision was not an unreasonable application of federal law, and federal habeas relief was due to be denied.
B. Witness Tucker
Anita Tucker also testified during the sentence phase. She told how she had met Brown while both of them were incarcerated in the Gwinnett County Jail. On one occasion, she and Brown were transported from that jail to the courthouse in the same car. During that trip, Brown had told her she “should play crazy and that [she] would not do any more than two years.” Tucker said that Brown knew of her because he had been incarcerated with her co-defendants in an unrelated case.
Tucker also testified about another meeting with Brown which had taken place after her conviction, when she was incarcerated at Hardwick Correctional Institution. This meeting occurred when Tucker was transported to Central State Hospital for a dental evaluation. While she was there, Tucker testified, Brown had asked her “what happened to you in Gwinnett County?” Tucker told Brown she “was found guilty and given a sentence of life plus sixty years,” to which Brown replied, “Well, I told you.” Tucker explained to the jury that his statement referred to their earlier conversation about “playing crazy.”
During her testimony, Tucker was asked whether she had received any benefit for her testimony, either for the testimony she was then giving or for her testimony at the first trial. Tucker answered by denying that she had received a benefit, explaining that she had been convicted of murder and armed robbery, and stating that she had served her sentence and period of parole. On cross-examination, Brown’s trial counsel inquired about how the prosecution had become aware of Tucker’s testimony. Tucker explained that Detective Burt Bla-nott, the investigator who worked her case, had come to see her at Hardwick prison before Brown’s first trial, and that was when she had told the detective about Brown’s statement.
Brown’s claim as it relates to Tucker is that his trial counsel should have discovered evidence that could have been used impeach her testimony. Specifically, in the state habeas proceedings, Brown contended that Tucker’s testimony could have been impeached on three grounds, none of which trial counsel had pursued: (1) Tucker and Brown were never at the dentist’s office on the same day; (2) Tucker had changed her story as to the location of her first conversation with Brown from a holding cell to the back seat of a patrol car; and (3) after the first trial, Tucker was treated favorably by the State in regard to her sentence.
The state habeas court concluded that trial counsel’s performance in regard to the credibility issues involving Tucker was deficient, because counsel could and should have discovered, and used to impeach Tucker, the information about her not being at the dentist’s office at the same time as Brown and about her arguably receiving favorable treatment in return for her testimony at the first trial. But the state court denied relief on the ineffective assistance claim relating to Tucker on prejudice [1313]*1313grounds. In this federal habeas proceeding, the district court agreed with the state habeas court that trial counsel’s representation was deficient, but found the state court’s conclusion that trial counsel’s representation did not result in prejudice to Brown’s defense was objectively reasonable.
C. Discussion and Analysis
We need not address the performance deficiency component of this ineffective assistance claim, because failure to satisfy the prejudice component is dispositive. See Strickland v. Washington, 466 U.S. 668, 697, 104 S.Ct. 2052, 2069, 80 L.Ed.2d 674 (1984) (“If it is easier to dispose of an ineffectiveness claim on the ground of lack of sufficient prejudice, which we expect will often be so, that course should be followed.”). The prejudice component requires Brown to establish that but for the deficient representation, there is a reasonable probability of a different result in the proceeding — here, the sentence proceeding. See Strickland, 466 U.S. 668, 694, 104 S.Ct. 2052, 2068, 80 L.Ed.2d 674 (1984). A reasonable probability is a probability sufficient to undermine confidence in the outcome — the outcome here, a death sentence.
Brown points out that use of mental illness as a mitigating circumstance was his central strategy in the sentencing phase, and that White and Tucker were important witnesses supporting the prosecution’s position that Brown was not mentally ill. It follows, he argues, there is a reasonable probability that if the impeachment evidence had been disclosed or discovered and used against White and Tucker, the jury would not have returned a verdict imposing a sentence of death. That is the issue the state courts faced but it is not the issue we must address, because our review of Brown’s claims is confined by 28 U.S.C. § 2254(d), as explicated by the Supreme Court in Williams v. Taylor, 529 U.S. 362, 120 S.Ct. 1495, 146 L.Ed.2d 389 (2000). Under § 2254(d)(1) and the Williams decision, we can grant relief only if we the state court decision denying relief is “contrary to” clearly established federal law or is an “unreasonable application” of federal law.
Brown does not dispute that the state habeas court correctly identified the controlling law, which is found in Strickland v. Washington, but he contends that the state court unreasonably applied Strickland’s prejudice prong. It is the objective reasonableness, not the correctness per se, of the state court decision that we are to decide. See Williams, 529 U.S. at 411, 120 S.Ct. at 1522 (“Under § 2254(d)(l)’s ‘unreasonable application’ clause, then, a federal habeas court may not issue the writ simply because that court concludes in its independent judgment that the relevant state-court decision applied clearly established federal law erroneously or incorrectly. Rather, that application must also be unreasonable.”) (O’Connor, J., concurring). We turn now to that issue.
The evidence of Brown’s guilt was overwhelming. At the sentence stage, Brown presented evidence of mental state mitigating circumstances, taking the position that he was seriously mentally ill. He called two expert witnesses to testify in support of the theory that he was schizophrenic, and he put forward some documentary evidence of his mental condition. The State presented an expert witness of its own whose testimony was that Brown was not schizophrenic. In other words, there was the usual disagreement of expert witnesses. Brown’s mother also testified to his mental problems and about his abusive childhood.
The testimony of White and Tucker hurt Brown’s chances at convincing the jury [1314]*1314that he was mentally ill, because it strongly suggested he had planned, and was faking, his mental illness defense. The effect the impeachment evidence would have had on the sentence verdict, however, is limited for two reasons. First, much of the evidence Brown touts as impeachment does not actually contradict the testimony of White and Tucker. That evidence, insofar as it relates to White, establishes only that White did not report the statement Brown made at booking to the prosecutor until asked six years after the fact whether Brown had acted normally that night. At no time during or before the trial did White ever claim that he had reported the statement any earlier. Moreover, the ha-beas court found as a fact that White was not lying about having heard the statement, and that is a factfinding which we must and do accept as presumptively correct. See 28 U.S.C. § 2254(e)(1).
Similarly, none of the impeachment evidence relating to Tucker directly contradicts her testimony at the retrial, which is where the death sentence under review originated. As the district court explained, Tucker’s statement to the police about her initial conversation with Brown is not necessarily inconsistent with any of her testimony during this retrial’s sentence hearing. Tucker’s testimony was that Brown had told her while they were in a deputy’s car that she should act insane. The evidence that Brown says could have been used to impeach that testimony is a report of what Tucker told the police during an earlier interview, namely, that Brown had made that statement to her “on at least one occasions [sic] when both she and James Willie Brown had been transported to court and were in the holding cell together.” The report’s second-hand recounting of Tucker’s statement to police is sufficiently ambiguous that it is not worth much as impeachment.
Brown also argues that at the first trial Tucker had testified that she later ran into Brown at the dental clinic on the grounds of Central State Hospital and that is where Brown, upon learning that she had been convicted, reminded her that he had told her to fake insanity. At this retrial, during the sentencing phase, Tucker testified that she encountered Brown on the Central State Hospital grounds, but she did not specifically identify the location as the dental clinic. At the habeas proceeding, Brown introduced evidence that supposedly showed he and Tucker were never at the dental clinic together. The impeachment value of that evidence is diminished by the fact that although it contradicted one detail of White’s testimony at the first trial, it did not conflict with any of her testimony at the retrial, and it is the result of the retrial that we have under review.
The second reason that the impeachment evidence aimed at the testimony of White and Tucker has limited effect is that evidence other than the testimony of White and Tucker also leads to the conclusion that Brown was faking his mental illness. For example, there was undisputed evidence that within a day or two after the crime Brown had sufficient possession of his mental faculties to have two tires on the car he used in the abduction changed. His doing so was powerful evidence of his consciousness of guilt, his awareness of the possibility he had left his automobile’s tire tracks near the scene of the crime out on the old logging road, and his efforts to avoid detection.
The aggravating circumstances were unequivocal and overwhelming. The jury found three statutory aggravating circumstances: (1) Brown had a prior felony conviction of armed robbery; (2) Brown committed the crime of rape during this murder; and (3) the capital crime Brown committed was outrageously or wantonly vile in that it involved both depravity of mind and torture to the victim before she [1315]*1315died. Not only that, but the jury also heard and was permitted to consider in sentencing Brown the fact that he had previously attacked two other women in a manner similar to the way he attacked the victim in this case, although those other two women had managed to escape with their lives. The totality of the evidence at sentencing weighs heavily against Brown on the prejudice issue. See Strickland, 466 U.S. at 695, 104 S.Ct. at 2069 (“In making this [prejudice] determination, a court hearing an ineffectiveness claim must consider the totality of the evidence before the judge or jury.”).
In view of all of these circumstances, we conclude the state habeas court’s decision that Brown had failed to establish the prejudice prong of his ineffective assistance claims is objectively reasonable within the meaning of § 2254(d).
II. THE BRADY CLAIMS
In addition to his ineffective assistance claims relating to the testimony of witnesses White and Tucker, Brown pursues Brady claims relating to them. He also presses a Brady claim relating to the potential testimony of an individual named John Wood, who did not testify at either of Brown’s trials.
A. Witnesses White and Tucker
Brown’s Brady claims relating to witnesses White and Tucker overlap with his ineffective assistance claim involving the impeachment evidence that might have been used against those two witnesses. He argues that if it was not deficient performance (and perhaps even if it was) for his counsel to fail to discover the impeachment evidence that could have been used against White and Tucker, then the State’s suppression of that evidence (or most of it) violated Brady. The state habeas court bypassed the suppression issue by holding that the evidence was not “material” for Brady purposes anyway. In this federal habeas proceeding, the district court also denied relief, but on different grounds with respect to the evidence relating to White and Tucker. As to White, the court concluded that the State had not suppressed the evidence, and as to Tucker, the court concluded that the evidence, although suppressed, was not material.
B. Witness Wood
Brown also contends that the State suppressed evidence relating to the corroboration of his alibi. The evidence in question was a note saying that an individual named John Wood had reported to the police that Brown’s brother had heard Brown come home at 12:30 a.m. on the night that Watson was killed.2 This information, Brown contends, would have led to the discovery of his brother, Johnny Brown, as a corroborating witness, which would have buttressed his defense at the guilt stage of his retrial that he was home at the time of the murder. Brown contends that this evidence would have made a difference at the guilt stage, and if not there then at the sentence phase where he could have used it to make a residual doubt argument.
The state habeas court denied relief on the ground that Brown could not show that the note was material. The court reasoned that this evidence would not have any chance in changing the outcome of Brown’s retrial because the State’s case against Brown was overwhelming. In this federal habeas proceeding, the district court also denied relief, but did so on the ground that Brown could not show that the prosecution had suppressed the note.
[1316]*1316C. Analysis and Discussion
We will assume for present purposes that the State suppressed the evidence relating to these three witnesses, which leaves the materiality of component of the Brady claim to be addressed. As with the ineffective assistance claims, we review for objective reasonableness, not per se correctness. See 28 U.S.C. § 2254(d). The prejudice component of an ineffective assistance claim and the materiality component of a Brady claim both require the same thing: the petitioner must establish that but for the deficient representation or suppression, there is a reasonable probability of a different result in the proceeding. Compare Strickland, 466 U.S. at 694, 104 S.Ct. at 2068 with Kyles v. Whitley, 514 U.S. 419, 433-34, 115 S.Ct. 1555, 1565, 131 L.Ed.2d 490 (1995). If the failure to use certain evidence does not result in prejudice for ineffective assistance purposes, the suppression of some of that same evidence will not be material for Brady purposes. With respect to the impeachment evidence relating to witnesses White and Tucker, that evidence, standing alone, is not material for Brady purposes for the same reasons we have explained that counsel’s failure to discover and use it was not prejudicial for ineffective assistance purposes.
But in making Brady materiality determinations, the collective impact of all of the suppressed evidence must be considered against the totality of the circumstances. See Kyles, 514 U.S. at 441, 115 S.Ct. at 1569. In other words, we must add to the effect of the impeachment evidence against White and Tucker any additional effect that the note relating to John Wood would have had on the result of the retrial.
At the retrial, Brown testified that on the night Watson was killed, he did not go with her to the Mark Inn Lounge, but went there alone, left alone, and was home by 12:30 a.m., which the defense argued was the earliest time that Watson could have been killed. Brown’s wife was going to tell the jury that Brown was home with her at 12:30 a.m., but on the day she was supposed to testify, she invoked the spousal privilege. Since Brown was not aware of any other witnesses who could testify that he was home at that time, his testimony, and defense, went uncorroborated.
The lack of corroboration hurt Brown’s chances of convincing the jury he was home at 12:30 a.m., but whether or not he was home then was not crucial to the prosecution’s case or to the jury’s verdict, either at the guilt stage or the sentence stage. As the district court explained, the exculpatory value of the note was limited because the time of death was never pinned down at trial. We review this finding of fact for clear error and find none. There was no evidence that definitively established the time of Watson’s death. The closest thing was the testimony of Dr. James Howard, a Forensic scientist with the Georgia Bureau of Investigation, who told the jury that based on the gastric contents and the rigor of the body when it was found, Watson died approximately two to two and a half hours after she ate her last meal.
When Watson ate her meal was never established with any certainty at trial. There was evidence that Brown and Watson arrived at the Mark Inn Lounge around 9:00 p.m., bringing with them at least some of their dinner in styrofoam containers. There is no evidence about how much of their dinner was in those containers, or when they first began eating that night. (Recall that Brown denied going there with Watson that night.) It is entirely consistent with all of the evidence that Watson could have finished her dinner by 9:30 p.m. and Brown could have killed her by 11:30 p.m. There is no evidence at [1317]*1317all that it would have taken Brown more than an hour to get from the murder scene to his home.
So, even assuming that the note about John Wood was suppressed, it adds nothing to the cumulative weight of the other allegedly suppressed evidence' — the impeachment evidence relating to Tucker and 'White. It follows that the state habeas court’s decision that Brown had failed to establish the materiality prong of his Brady claims is objectively reasonable within the meaning of § 2254(d).
III. THE GIGLIO CLAIM
Finally, Brown also contends there was a Giglio violation relating to Tucker’s testimony at the sentence phase. In order to prevail on a Giglio claim, Brown must establish that the prosecutor “knowingly used perjured testimony, or failed to correct what he subsequently learned was false testimony,” United States v. Alzate, 47 F.3d 1103, 1110 (11th Cir.1995), and that the falsehood was material. See Giglio v. United States., 405 U.S. 150, 92 S.Ct. 763, 31 L.Ed.2d 104 (1972); United States v. Agurs, 427 U.S. 97, 103, 96 S.Ct. 2392, 2397, 49 L.Ed.2d 342 (1976). The materiality prong is easier to establish with Giglio claims than with Brady claims. For Giglio purposes, “the falsehood is deemed to be material ‘if there is any reasonable likelihood that the false testimony could have affected the judgment of the jury.’ ” Alzate, 47 F.3d at 1110 (quoting Agurs, 427 U.S. at 103, 96 S.Ct. at 2397).
During the sentencing phase of the retrial, Tucker testified that she received no benefit in return for her testimony against Brown, either in the first trial or in the retrial. Brown’s Giglio claim is based upon his contention that the above statement was false and that the State knew it. The State concedes that the prosecutor took steps to help Tucker get transferred to a halfway house, but it denies that he did so because Tucker had testified in this case. Instead, the State’s position is that the prosecutor did that in return for Tucker’s cooperation with authorities against her own co-defendants in an unrelated case.
Brown offers several documents which he contends show that Tucker lied on the witness stand and that the prosecution knew it. He points to letters Tucker wrote and phone calls Tucker made to the prosecutor’s office asking for help, including help in the form of a transfer. He further points to documents of the Georgia Board of Pardon and Paroles and the Department of Corrections indicating that the prosecutor talked with the Department of Corrections about Tucker being assigned to work release. There is also the fact that the prosecutor intervened to help Brown about eight months after Brown’s first trial. From those documents Brown infers that the prosecutor did help Brown because of her testimony against him.
The state habeas court found as a fact that Brown had not shown that Tucker had lied when she denied receiving favorable treatment in return for her testimony at Brown’s first trial. The state court found no evidence in the parole board documents, which are the centerpiece of Tucker’s claim, that the benefit she had received was connected to her testimony in this case instead of to her cooperation against her own co-defendants in another murder case. At another point in its decision, the state habeas court also found that the timing of Tucker’s transfer did not show that there was any inappropriate conduct between Tucker and the prosecutor surrounding her testimony. In this federal habeas proceeding, the district court agreed with the state court and found that Brown had failed to show that the testimony of Tucker about receiving no [1318]*1318benefits for her testimony against Brown was false. It followed that the state court’s denial of relief was objectively reasonable.
We review the state court’s conclusion that there was no Giglio violation for objective reasonableness, not per se correctness. See 28 U.S.C, § 2254(d). We take the state habeas court’s factual finding that there was no inappropriate conduct between Tucker and the prosecutor surrounding her testimony as presumptively correct.
The essence of Brown’s contention is that the timing of the prosecutor’s assistance can only lead to the conclusion that he assisted in response to Tucker’s testimony against Brown. Yet the very document upon which Brown places so much reliance, the memorandum evidencing the prosecutor’s communication on behalf of Tucker with the Georgia Department of Corrections, clearly states that the prosecutor was intervening because Tucker had been helpful in the prosecution of her co-defendants in that murder case. Brown’s speculation that the timing suggests, notwithstanding what the key document says, that the prosecutor actually intervened because of Tucker’s testimony against Brown is not enough to carry his burden of rebutting by clear and convincing evidence the state court’s contrary factfinding. See 28 U.S.C. § 2254(e). The state court’s decision of this claim was not objectively unreasonable.
IV. CONCLUSION
The district court’s judgment denying the petition for writ of habeas corpus is AFFIRMED.