I'm struggling a bit on how to construct a proof for this question.
Suppose I have a Lipschitz function $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfies $|f(x)-f(y)|\leq L|x-y|$ which is differentiable at some point $c\in\mathbb{R}$. Then $f'(c)\leq L$.
I understand the idea behind it. I know that
$$\lim_{x\to c}\frac{f(x)-f(c)}{x-c}$$
exists and is finite. I also know that
$$\frac{f(x)-f(c)}{x-c}\leq \left|\frac{f(x)-f(c)}{x-c}\right|\leq L$$
and intuitively it makes sense to just take limits of the left-most and right-most term to show that $f'(c)\leq L,$ but why am I allowed to do that? And if it's not possible for me to try that how else would I attempt to prove this?
From $ |\frac{f(x)-f(c)}{x-c}|\leq L$ we get with $x \to c$ that
$|f'(c)| \le L$, hence
$$ f'(c) \le |f'(c)| \le L.$$