Even to the Jews who took the prophets seriously, it needed to be shown that the death and resurrection of the Messiah were predicted by the prophets — and that Jesus had fulfilled not just some of the messianic prophecies but all of them.
The One whom David foresaw in Psalm 16:10 is the only person in the world who, having been killed, did not “see corruption.” His resurrection proved Him to be the Son of God. It proved everything about the gospel to be true. Therein lies our hope.
Although God might have let justice run its course, He was not willing to leave us lost. If we had ruined the perfection He had created us to enjoy, He would allow us to come back to Him — even if He had to take the punishment for our sin.
If we’ve obeyed the gospel, we are not any more alive physically, but we certainly possess a spiritual life we didn’t used to have. When we were baptized, God “raised us up with [Christ] and seated us with him in the heavenly places in Christ Jesus.”
Sin is not a problem we can work our own way out of. Having alienated ourselves from God, we were absolutely helpless. What we needed was not simply encouragement or a better moral example. We needed a sinless Savior who would die on our behalf.
What began to “go forth” from Jerusalem on Pentecost was the preaching of the truth about God’s means of saving the world in Christ. Since then, anyone who has heard that message — and obeyed it — has come into a forgiven relationship with God.