Did Jesus Exist?

It is very easy to dismiss the facts and think a man called Jesus never lived around 2000 years ago. So what are the historical facts that prove he did. Firstly, Christians know he existed because the bible says so. However, if you don't believe in Jesus then you won't believe the bible is correct. Therefore I'm not going to use the bible as my source of fact.
 
Fact 1: Roman historian Carius Cornelius Tacitus
Tacitus lived between AD 56 and AD 117 and was a senator of the Roman Empire as well as a Roman Historian. The wrote a number of works but the Annals and the Histories examine Roman Emperors Tiberius, Claudius and Nero. The Annals was Tacitus's final work and book 15 written around AD 116 talks about Christus as a person convicted by Pontius Pilate during Tiberius' reign.

"...Nero fastened the guilt and inflicted the most exquisite tortures on a class hated for their abominations, called Christians by the populace. 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...

Fact 2: Jewish Historian Flavius Josephus
Josephus lived around AD37 and AD 100 was a Jewish Historian and hagiographer (study of saints). His writings covers events like the First Jewish-Roman War which resulted in the Destruction of Jerusalem in AD 70. However, Antiquities of the Jews (c. 94) talks about a man called Jesus.

"Now there was about this time Jesus, a wise man, if it be lawful to call him a man; for he was a doer of wonderful works, a teacher of such men as receive the truth with pleasure. He drew over to him both many of the Jews and many of the Gentiles. He was [the] Christ. And when Pilate, at the suggestion of the principal men amongst us, had condemned him to the cross, those that loved him at the first did not forsake him; for he appeared to them alive again the third day; as the divine prophets had foretold these and ten thousand other wonderful things concerning him. And the tribe of Christians, so named from him, are not extinct at this day."