Show there exists a sequence of positive real numbers s.t. ...

Fact 1: If $f:[0,1]\to\mathbb R$ is measurable and $|f(x)|$ is finite for almost every $x$, then for any $\epsilon>0$ there is a number $M$ such that the set $\{x:|f(x)|>M\}$ has measure at most $\epsilon$.

Reason: Since the intersection $\bigcap_{k=1}^\infty \{x:|f(x)|>k\}$ has measure $0$ and the sets in this intersection are nested, it follows that $m[\{x:|f(x)|>k\}]\to 0$ as $k\to\infty$.

Choose $c_n$ on the basis of Fact 1, so that $m[\{x:|f(x)|/c_n>1/n\}]<2^{-n}$.

The rest is an application of Borel-Cantelli: the set of number $x$ such that $|f(x)|/c_n>1/n$ infinitely often has measure zero. Rephrase this as: for almost every $x$ we have $|f(x)|/c_n\le 1/n$ for all sufficiently large $n$.