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I have to show that $|f '(x)| \leq 1, \ \forall x\in R$. The information I have been given is $|f(x)-f(y)|\leq |x-y|$ ... cauchy schwarz inequality. This is for calculus. Thanks so much.

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Use the definition of $f'(x)$: $$ f'(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}{h} $$ Now, take the absolute value of both sides: $$ |f'(x)|= \lim_{h\to 0}\left|\frac{f(x+h)-f(x)}{h}\right|=\lim_{h\to 0}\frac{|f(x+h)-f(x)|}{h} $$ Using the given information, what do you know about $|f(x+h)-f(x)|$?

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