3 to whom He also presented Himself alive after His suffering by many infallible proofs, being seen by them during forty days and speaking of the things pertaining to the kingdom of God.
3 to whom He also presented Himself alive after His suffering [Crucifixion] by many infallible [convincing, decisive] proofs,
The “many infallible proofs” are the events after Jesus’ Resurrection, covering 40 days (1 Jn 1:1). The only time the word “proofs” occurs in the New Testament is in this verse.
The word for “proofs” is a technical term from logic carrying the idea of demonstrative evidence. “Infallible proofs” show demonstrable evidence rather than that provided by witnesses. The apostles would never have risked their lives for a dead Christ; they needed to know for sure that He was raised from the dead. The apostles became convinced without question that He rose physically from the dead.
being seen by them during forty days
The Lord showed himself many times during the post-Resurrection period and before the Ascension. The reality of their visual evidence of His Resurrection by His personal post-Resurrection appearances was the confirmation of their faith (1 Co 15:3-8).
and speaking of the things pertaining to the kingdom of God.
Jesus was the rejected King. Israel said, “We will not have this man reign over us.” They had “no king but Caesar.” Jesus spoke of the “kingdom of God”; that is, He turned their thinking to the Great Commission, where He charged His people to take the gospel to every creature (Lu 24:45-47). The “kingdom of God” here is the universal kingdom, not the Millennial kingdom.
Christianity rests its case on proven facts.
The Resurrection of Christ and its surrounding events is the foundation of Christianity. The post-Resurrection occurrences of Christ attest to the validity of His Resurrection. Christianity bases its case on facts, truth, and evidence. The apostles rested their case on what they saw and evidenced (1 Jn 1:1). Peter made it very clear that the apostles did not base their evidence on myth (2 Pe 1:16). They knew the Lord before and after His Crucifixion and Resurrection. Thomas declared that he would not believe unless he saw the “proof” himself. After seeing the proof, he declared, “My Lord and my God” (Jn 20:28).