Suppose a function $f$ is in both $L^1(\mathbb{R})$ and $L^2(\mathbb{R})$. For $m=1,2,\ldots$, let $$f_m(x) = f(x)\cdot \chi_{[-m,m]}(x)\cdot \chi_{\{ \lvert f(y)\rvert \leqslant m\}}(x)$$ where $\chi$ is the characteristic function. Then why is $f_m(x)\rightarrow f(x)$ in both $L^1$ and $L^2$?
This was used in an answer to this post, but I don't understand the explanation there. Perhaps someone can explain it in simpler terms, or show the proof?