# Differential Inequalities involving Absolute Values

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.

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 $|f(x+h)-f(x)|$?