Timothy Gowers is a Fields Medal winner who writes a blog about math. He has written a post asking “When is proof by contradiction necessary?”

“This is a question that arises when I am teaching somebody who comes up with a proof like this. “Suppose that the sequence is not convergent. Then … a few lines of calculation … which implies that . Contradiction.” They are sometimes quite surprised when you point out that the first and last lines of this proof can be crossed out. Slightly less laughable is a proof that is more like this. “We know that . Suppose that . Since the derivative of has absolute value at most 1 everywhere, it follows that , which is a contradiction. Therefore, .” There, it is clearly better to work directly from the premise that via the lemma that to the conclusion that . However, the usual proof of the lemma *does* use contradiction: one assumes that the conclusion is false and applies the mean value theorem.”

Source: Gowers’s Webblog