I'm trying to prove the following theorem:
Let $f:\mathbb{R}\to\mathbb{R}$ be a differentiable function, then $Y\subset\mathbb{R} \text { negligible}\Rightarrow f(Y)\subset\mathbb{R} \text{ negligible}$.
Using the following theorem(*):
Let $Y\subset\mathbb{R}$ be Lebesgue-integrable and $f:\mathbb{R}\to\mathbb{R}$ a function s.t. $\forall y\in Y$ f differentiable at $y$ and $|f'(y)|<n$, then $\exists B\subset\mathbb{R}$ Borel s.t. $f(Y)\subset B$ and $\lambda(B)\leq n\lambda(Y)$.
My proof is as follows:
Let $Y\subset\mathbb{R}$ be negligible and define $Y_{n}\subset Y$ by $Y_{n}:=\{y\in Y:|f'(y)|<n\}$, then by applying theorem (*)$\ \exists B_{n}\subset \mathbb{R}$ Borel s.t. $f(Y_{n})\subset B_{n}$ and $\lambda(B_{n}) \leq n\lambda(Y_{n}) = n*0 = 0 \ \forall n\in\mathbb{N}$, thus $\lambda(f(Y_{n}))\leq\lambda(B_{n})\leq 0$, so that $\lambda(f(Y)) \leq \sum \lambda(f(Y_{n})) = \sum 0 = 0$ thus $f(Y)$ is negligible.
However, the hint in the exercise says it's not as easy as it looks. So I'm wondering: assuming my proof is too short, what's wrong?