Question applying Mean Value Theorem Question:
Suppose $f : \mathbb R \to \mathbb R$ satisfies $|f'(x)|\leq 1 $ for all $x \in \mathbb R$, show that 
$$|f(x) - f(y)| ≤ |x-y|$$
for all $x,y \in \mathbb R$.
Hello I have this question from my differential analysis class and I'm very stuck on how to answer it.
Attempt:
I have a feeling that it has to do with the mean value theorem although I'm not sure how to use the information given about the derivative of $f$ to answer the question at all. 
I've only got so far as to write down $-1≤f'(x)≤1$ and $f'(c) = 1$ and have no idea at all how I'm going to tie it to the inequality involving $x$ and $y$.

Hint: You can write: $$f(x)-f(y) = f'(c)(x-y) \implies |f(x)-f(y)| = |f'(c)||x-y|,$$ for some $c$ between $x$ and $y$. And $|f'(c)| \leq 1,$ so...
