If $f$ is a $K$-lipschitz function, and if $A \subset \mathbb{R}$, then is $diam(f(A))\leq Kdiam(A)$?

Question

$A$ being of form $a,b \in \mathbb{R}, A= [a,b]$. Would that be true? I have no idea if it's true, even less how to prove it. Or would there be a similar inequality?

Answer 1

Let $A$ be a bounded subset of $ \mathbb R$ and $f:A \to \mathbb R$ a $K$-lipschitz fuction.

For $x,y \in A$ we have $|x-y| \le diam(A)$, hence

$|f(x)-f(y)| \le K|x-y| \le K diam (A)$.

This gives $diam(f(A))\leq Kdiam(A)$