# Getting to know your math tools.

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 $(a_n)$ is not convergent. Then … a few lines of calculation … which implies that $a_n\rightarrow a$. 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 $|y-x|<\delta$. Suppose that $|\sin(y)-\sin(x)|\geq\delta$. Since the derivative of $\sin$ has absolute value at most 1 everywhere, it follows that $|y-x|\geq\delta$, which is a contradiction. Therefore, $|\sin(x)-\sin(y)|<\delta$.” There, it is clearly better to work directly from the premise that $|y-x|<\delta$ via the lemma that $|f(x)-f(y)|\leq\|f'\|_\infty |x-y|$ to the conclusion that $|\sin(y)-\sin(x)|<\delta$. 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