Is the image of a Null set is still a Null set under a differentiable function whose derivative is bounded?

Question

Given $f: \mathbb{R}\rightarrow\mathbb{R}$ is a differentiable function and its derivative is bounded. If $A$ is a null set, is $f(A)$ also null?

We just start real analysis. There are not many things we can use. I found this but I think there should be a way to done it just by using the definition of the null set. My idea is that we might relate the $\epsilon - \delta$ of definition differentiable with the $\epsilon$ in the definition of null set. But it seems I get problem on it

Answer 1

Null set means you can cover it with intervals of arbitrarily small total length. Bounded derivative implies that the function is Lipschitz, so the length of the image of any interval is at most $C$ times the length of the original interval for some fixed $C>0$. Combine these two observations, and you are done.