Why should we prove obvious things? Obviously, there are obvious things in mathematics. Why we should prove them?


*

*Prove that $\lim\limits_{n\to\infty}\dfrac{1}{n}=0$?

*Prove that $f(x)=x$ is continuous on $\mathbb{R}$?

*$\dotsc$


Just to list few examples.

In the 19th century it took a while to clarify the notion of continuity of a function.  Ampère actually published a "proof" that any continuous function is differentiable, period. Even Cauchy seems to have felt that a continuous function should have at most a finite number of points of discontinuity (Boyer in his book erroneously reverses the roles of Ampère and Cauchy). For a more detailed discussion see this article. Such a "theorem" must have seemed intuitively "obvious" at the time.
