Question

Why did Christopher Columbus not believe the earth was not flat and how did he prove to the people it wasn't?

Answer 1

Let's tell the actual truth. Everyone believed the Earth was round. But Christopher Columbus believed it was bigger. 

Answer 2

 Some have already noted that Columbus's voyages COULD not have "proved" the world round --only circumnavigating the globe would do that in the sense you are thinking of. 

But the more important question is, "Did Columbus INTEND to prove the world round (by sailing west to reach the Far East)?" 

To that the answer is a decided "NO!!" 

The notion that Columbus was opposed by churchmen who thought the world flat is a 19th century myth, popularized by Washington Irving's popular but very poorly researched "biography" of Columbus, and the work of some strongly anti-Christian writers (two in particular). 

The truth is - The churchmen in the story were CORRECT about the size of the earth (and that there was more water than land, contra Columbus's other error), which would have made the journey Columbus proposed a bad INVESTMENT.