# Sequence of measurable functions $f_n$ on $\mathbb R^n$ converges to $0$ in measure

I am having problems on how to start, any hints are appreciated. Thank you.

Show that a sequence of measurable functions $f_n$ on $\mathbb R^n$ converges to $0$ in measure if and only if $$\lim_{n \to \infty}\left(\int \frac{|f_n|}{1+|f_n|}\right)=0.$$

Hint: apply Markov's inequality. For any $\epsilon \in (0,1)$, we have $$\int \frac{|f_n|}{1 + |f_n|} \geq \epsilon\cdot \mu\left\{ \frac{|f_n|}{1 + |f_n|} \geq \epsilon\right\}$$ Noting that $$\frac{|f_n|}{1 + |f_n|} \geq \epsilon \iff |f_n| \geq \frac{\epsilon}{1-\epsilon}$$