Show there exists a sequence of postive real numbers s.t. [duplicate]

Let $(f_n)_{n\in\mathbb N}$ be a sequence of measurable functions on $[0,1]$ with $\forall n\in\mathbb N, \lvert f_n(x) \rvert < +\infty$ a.e. Show there exists a sequence $(c_n)_{n\in\mathbb N}$ of positive real numbers s.t. $f_n(x)/c_n \rightarrow 0$ for almost every $x$ in $[0,1]$. Hint: use the borel-cantelli lemma- pick the sequence $c_n$ so that $$m[x \in [0,1]:\frac{|f_n(x)|}{c_n}>1/n]<1/2^n$$ where m is the measure.