Paul publicly maintained that Peter believed Jesus resurrected.1 This is relevant because Paul knew whether Peter actually believed this (having interacted with him on several occasions, and staying with him for fourteen days).
Acts reports that Peter proclaimed Jesus's resurrection.1 This is relevant because the Acts reports here are reliable. (See: Speeches in Acts are reliable)
The Jerusalem church publicly maintained that Jesus resurrected. This is relevant because Peter was a member of the Jerusalem church. (In fact, he was a leader in it.)1
1 Peter 1:21 says “God, who raised Him from the dead,” and 1 Peter 3:21 says “through the resurrection of Jesus Christ.” This is relevant because 1 Peter was written by Peter the Apostle.1
1 Corinthians 15 reports “that He was raised, and that He appeared to Cephas who is Peter, _then to the twelve.” This is relevant because 1 Corinthians was written by Paul (who intimately knew Peter's official position, saying that Jesus appeared to Peter in v.3).
Moreover, the content in v3-5 is material that was already in circulation, Paul was “passing it down”, and notes that “Whether then it was I or they [Peter et al.], so we preach and so you believed.” In other words, the message of Jesus's appearance to Peter et al. was preached by Peter.
All the 12 apostles (minus Judas) maintained that Jesus resurrected. [Forthcoming] This is relevant because Peter was one of the 12 apostles.
Peter believed that Jesus made resurrection appearances to him. Four evidences:
• …the A.D. 30 Jerusalem church said so
• …the Corinthian church said so
• …Paul said so
• …Luke said so
This is relevant, of course, because it entails that he believed Jesus was alive again from the dead (and far less unlikely that he believed Jesus resurrected in particular).
Peter was marytred. This is relevant because it is best explained by his persistent proclaiming Jesus's resurrection (something controversial).