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 $y$.

You're on the good way: by the mean value theorem and for $x<y$ there's $c\in(x,y)$ such that
$$f(x)-f(y)=f'(c)(x-y)\implies |f(x)-f(y)|=|f'(c)||x-y|\le |x-y|$$
