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


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