Bounded derivative implies Lipschitz
This is a very common proof, but there is one point, which many just seem to take for granted.
At one point I'll use the mean value theorem to say
$$ w.l.o.g.\quad x > y \quad x,y \in (a,b) $$
And from the mean value theorem I get a term like this
$$\frac{f(x)-f(y)}{x-y} = f'(\xi) \leq L $$
But people often conclude $$|f(x)-f(y)| \leq L|x-y|$$
Is that step always true? i.e. is it always true that
$$ x = y \rightarrow |x|=|y| $$
The mapping $x\mapsto |x|$ is a function like any other, and for any function $f$, it is true that if $a=b$, then $f(a)=f(b)$.