Nothing in history can be proven the way we can prove something in a laboratory. However, the resurrection of Jesus is a historical fact much more fully attested to than most other events of ancient history we take for granted. Every effort to account for the birth of the church apart from Jesus's resurrection flies in the face of what we know about first-century history and culture. If you don't short-circuit the process with the philosophical bias against the possibility of miracle, the resurrection of Jesus Jesus has the most evidence for it.
The problem is, however, that people do short-circuit the investigation. Instead of doing the work of answering these very tough historical questions and then following the answers where they lead, they bail out with the objection that miracles are impossible. N.T. Wright makes a scathing response:
The early Christians did not invent the empty tomb and the meetings or sighting of the risen Jesus... Nobody was expecting this kind of thing; no kind of conversion experience would have invented it, no matter how guilty (or how forgiven) they felt, no matter how many hours they pored over the scriptures. To suggest otherwise is to stop doing history and enter into a fantasy world of our own.
I sympathize with the person who says, "so what if I can't think of an alternate explanation? The resurrection just couldn't happen." Let's not forget, however, that first-century people felt exactly the same way. They found the resurrection just as inconceivable as you do. The only way anyone embraced the resurrection back then was by letting the evidence challenge and change their worldview, their view of what was possible. They had just as much trouble with the claims of the resurrection as you, yet the evidence- both of the eyewitness accounts and the changed lives of Christ's followers- was overwhelming.
Each year at Easter I get to preach on the Resurrection. In my sermon I always say to my skeptical, secular friends that, even if they can't believe in the resurrection, they should want it to be true. Most of them care deeply about justice for the poor, alleviating hunger and disease, and caring for the environment. Yet many of them believe that the material world was caused by accident and that the world and everything in it will eventually simply burn up in the death of the sun. They find it discouraging that so few people care about justice without realizing that their own worldview undermines any motivation to make the world a better place. Why sacrifice for the needs of others if in the end nothing we do will make any difference? If the resurrection of Jesus happened, however, that means there's infinite hope and reason to pour ourselves out for the needs of the world. In a sermon, N.T. Wright said:
The message of the resurrection is that this world matters! That the injustices and pains of this present world must now be addressed with the news that healing, justice, and love have won... If Easter means Jesus Christ is only raised in a spiritual sense - [then] it is only about me, and finding a new dimension in my personal spiritual life. But if Jesus Christ is truly risen from the dead, Christianity becomes good news for the whole world- news which warms our hearts precisely because it isn't just about warming hearts. Easter means that in a world where injustice, violence and degradation are endemic, God is not prepared to tolerate such things- and that we will work and plan, with all the energy of God, to implement victory of Jesus over them all. Take away Easter and Karl Marx was probably right to accuse Christianity of ignoring problems of the material world. Take it away and Freud was probably right to say Christianity is wish fulfillment. Take his away and the Nietzsche probably was right to say it was for wimps.
Excerpt from The Reason for God: Belief in an Age of Skepticism by Timothy Keller.