Josephus says he was seen alive
Josephus was a Jewish historian writing for a Roman audience around AD 93. He wasn't a Christian. In his history of the Jewish people, he recorded Jesus' execution and what followed:
"And when, upon the accusation of the principal men among us, Pilate had condemned him to a cross, those who had first come to love him did not cease. He appeared to them spending a third day restored to life, for the prophets of God had foretold these things and a thousand other marvels about him."
Crucified under Pilate. Seen alive by his followers on the third day. Written down by a Jewish historian, in Rome, sixty years later.
Tacitus says it broke out again
Tacitus, the Roman historian, despised the Christians. But look at what he says happened after Jesus was executed:
"Christus, from whom the name had its origin, suffered the extreme penalty during the reign of Tiberius at the hands of one of our procurators, Pontius Pilatus, and a most mischievous superstition, thus checked for the moment, again broke out not only in Judæa, the first source of the evil, but even in Rome."
Rome executed the leader, and the movement stopped. Then it came back, in Judea first, where it had died, and then all the way to Rome.
Tacitus doesn't explain why. The apostles did: they said they had seen him alive.
His enemies said the body was stolen
Here is the detail that matters most. Jesus' opponents had every reason to end the whole thing. All they needed to do was point to the tomb, or produce the body.
They never did. Instead, according to Matthew, the chief priests paid the guards to say this:
"Say you, that his disciples came by night, and stole him away when we were asleep... So they, taking the money, did as they were taught. And this word was spread abroad among the Jews even unto this day."
Matthew 28:13 and 15
And it wasn't only Matthew who heard it. Around AD 155, more than a century later, Justin Martyr wrote to a Jewish teacher named Trypho and said that his people had "sent chosen and ordained men throughout all the world" to say that "his disciples stole him by night from the tomb."
"The disciples stole the body" is an admission. You only need to explain why a tomb is empty if it is empty.
For a hundred years, the opponents of Christianity agreed on one thing: the body was gone.
"Check the Roman records"
The Romans kept records, and the early Christians knew it.
Around AD 155, Justin Martyr wrote a defence of Christianity addressed to the Emperor Antoninus Pius himself. Describing Jesus' miracles and crucifixion, he told the emperor:
"And that He did those things, you can learn from the Acts of Pontius Pilate."
Around AD 197, Tertullian went further. He wrote that Pilate had reported to the Emperor Tiberius about Jesus, and that Tiberius took the matter to the Senate. The Church historian Eusebius later recorded the same account: Pilate "informed Tiberius of the reports which were noised abroad through all Palestine concerning the resurrection."
Pilate's reports haven't survived. Very little Roman provincial paperwork from that time has. But Justin and Tertullian weren't writing to fellow believers. They were writing to Roman rulers, who had the archives, and telling them to go and look.
The witnesses
Paul, writing around AD 55, lists the people who saw the risen Jesus: Peter, the twelve, "more than five hundred brethren at once: of whom many remain until this present," then James, then all the apostles (1 Corinthians 15:5 to 7).
Many of them were still alive when he wrote it. Anyone could have gone and asked them.
Why this is a big deal
A Jewish historian recorded that Jesus was seen alive on the third day. A Roman historian recorded that the movement, crushed at his death, broke out again. His enemies spent a century explaining why the tomb was empty, and never once claimed it wasn't. And his followers told the Roman emperor to check Pilate's own reports.
Nobody outside the Bible ever produced the body.