Yes, we can! The same way we prove other ancient people existed.
1…we have reliable accounts of secular writers & historians that a man named Jesus lived in 1st century Israel. Josephus, Pliny the Younger, Cornelius Tacitus, Suetonis, Lucian of Samosata, Mara Ben-Serapion, Thallus, Phlegon, Celsus, many Jewish rabbis and the Jewish Talmud (hardly an unbiased source) and many Gnostic documents testify to His existence.
Josephus, a Jewish priest, general & historian, wrote in his Antiquities of the Jews that “About the time there lived Jesus, a wise man, if indeed one ought to call him a man…” He also referred to “James, the brother of Jesus who is called the Christ.”
Cornelius Tacitus, considered the most reliable of ancient historians, wrote of Jesus’ death as historical fact when he said, “Christus, the founder of the name (Christians) was put to death by Pontius Pilate…”
Mara Ben-Serapion, writing to encourage his son to pursue wisdom, referred to the lives of Socrates, Pythagoras and Christ as historical people who paid a price for their wisdom.
The Jewish Talmud and other writings from Jewish Rabbis spoke of Jesus in several places.
Almost all of the above sources were very antagonistic to Jesus & Christians.
By contrast, there are only 9 non-Christian sources who mention the Roman Emperor Tiberias Ceasar.
2…we have reliable evidence from the New Testament which records the life and teachings of Jesus. If you doubt the accuracy of the New Testament, take my class “How We Got the Bible” om this website. Historically, it is the most accurate of any ancient document in existence.
3… it is absurd to believe that thousands of people in the 1st century, many of whom knew Jesus personally, would devote themselves to a person who NEVER existed. Many of these Christ-followers were persecuted not just by governments but family and friends also. Many of them died for their belief in Jesus. How many people would die for a person who NEVER existed?
4…Without question, almost all scholars and historians agree that Jesus existed. Even Dr. Bart Ehrman, the agnostic department chairman of Religion at the University of North Carolina at Chapel Hill said that no reputable scholar disputes that He existed and lived in 1st century Israel.
5…Remember that Jesus never traveled more than a few miles from His hometown in His entire life. He also only had 3 years of public ministry, not a lifetime of 40-50 years. Historically speaking, we shouldn’t know anything about Him at all. The fact that we know so much about Him attests to His existence.
“That’s Just Your Interpretation” by Dr. Paul Copan, page 101
“A Ready Defense” by Josh McDowell, page 191