The book of Acts begins with Luke referring to his account of the gospel, which had also been addressed to "Theophilus," meaning "lover of God." The fact that Luke calls him "most excellent" in Luke 1:3 has caused some to suggest that Theophilus was a high government official (e.g., Acts 23:26; 24:3; 26:25). Acts could have been written to convert him or to confirm him in his faith, if he already was a Christian. The book of Luke was a record of Jesus' good works and teachings up through His ascension. This first chapter of Acts is transitional in that it begins immediately before His ascension and proceeds on from there.
Before His ascension, Jesus gave "commandments to the apostles whom He had chosen." Two commandments were recorded on that occasion: (1) "Repentance and remission of sins should be preached in His name to all nations, beginning at Jerusalem" (Luke 24:47) and (2) "Tarry in the city of Jerusalem until you are endued with power from on high" (24:49). This "power from on high" would be received via the baptism with the Spirit, which would then fully equip the apostles to preach the gospel to all peoples, beginning at Jerusalem. It is worth noting that not just anyone could become an apostle of Christ--he had to be chosen for the task (as well as meet other criteria; cf. Acts 1:21,22).
After the Lord was crucified and buried, He arose from the dead on the third day in accordance with prophecy (cf. I Cor. 15:3,4). He then "presented Himself alive" a number of times after His resurrection (cf. I Cor. 15:5-8) "by many infallible proofs" (Acts 1:3). To claim a proof as infallible is to affirm that it is reliable, convincing, without error, and unable to be overthrown logically. How did Jesus prove Himself to be risen from the dead in a way that cannot be denied logically? Here are several aspects to contemplate:
Jesus' resurrection was unexpected by His followers (cf. Luke 24:1-4; John 20:25). Thus, there was no delusion which resulted from any expectation of seeing Him. Since they weren't looking forward to His resurrection (though they should have been!), certainly they could not have all imagined Him in the various places and times He appeared over a span of forty days (before ascending to heaven). He ate with them, talked with them, taught them more about the kingdom, allowed them to touch Him, worked miracles, and showed Himself to be the same friend with whom they were so well acquainted. On one occasion He appeared to over 500 people at one time! Perhaps a few could be deceived or deluded, but not this many! Additionally, if there was any deception (which there is no evidence of), forty days would have provided plenty of time to uncover it as well as identify whoever was "pretending" to be the risen Lord.
Jesus' dead body was nowhere to be found. This is a significant proof. If Jesus did not rise from the dead, where was the body? If the enemies of Jesus had it in their possession, certainly they would have produced it to squelch Christianity as it grew rapidly in Jerusalem (e.g., Acts 2ff). If the disciples of Christ had His dead body in their possession, then they flawlessly executed the most devious deception our world has ever known! We can discount that possibility, however, with the following reason.
The change in behavior of the apostles proves they believed Jesus was risen from the dead. Obviously, they would not have believed this had they hidden or destroyed Jesus' dead body in some way. The fact that they were willing to suffer tremendous persecution and loss for the sake of Christ is an infallible proof that they knew He was alive! Rational people will only carry a deception so far before confessing their fraudulent behavior in order to escape a serious punishment (like execution). The apostles were willing to die for their faith, and for what gain physically? No one in their right mind would give up everything for that which they knew was a lie! Thus, the apostles knew Jesus really was risen from the dead! Their subsequent behavior shows that they were not crazy but behaving as men who were firmly convinced and fully dedicated to their cause.
Praise be to God that the resurrection of Christ has been proved infallibly, without which Christianity would fall (cf. I Cor. 15:14-19)!