Jesus is a merciful and compassionate God and he loves Peter. As such, he would like peter to know what he has said to him is true....
He got to talk privately to Peter because Peter was the first disciple to his tomb.
Jesus.
his disciples, before Jesus ascended into heaven
Jesus did not ever appear to Peter privately after His resurrection.
Peter was one of Jesus' closest friends and felt most comfortable with him.
There are several proofs that Jesus is alive: the most simple proof was when He appeared to His disciples, and showed them His wounds. Then He appeared before His disciples (and hundreds of others) for 40 days. Jesus is still alive because He is still doing a lot of miracles for a lot of people and curing them.
Jesus appeared to the disciples in the room, he also met two disciples on the road to Ema mus.
It was the angels.
fish baked on charcoal.
Jesus' disciples
The sea of Galilee
[John 20:24] Thomas called Didymus aka Doubting Thomas because when the disciples told him they had seen Jesus, Thomas would not believe them unless he had tangible proof which he got 8 days later when Jesus reappeared.
After he died he resurrected, appeared to the disciples before ascending to heaven. His resurrection meant he overcame death.