Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.
Sign up to join this community
Graphically, it is easy to see but to analytically derive the following inequality,
$$|e^x-e^y| \le |x-y| \ \ \ \ \forall x,y \in (-\infty,0] \ \ \ \ (1)$$
do I just apply the MVT and show that the derivative of at a certain c between x and y equals $0$ hence the ratio is less than 1 and thus we derive $(1)$
The idea is that $e^x$ is bounded on this interval since it is monotonically increasing.
By MVT, we know there exists some $c \in (-\infty,0)$ such that $$|e^x - e^y| = |e^c||x-y|$$ for any $x,y \in (-\infty,0]$. But notice that $e^c$ is bounded above by 1. This gives the desired result.
