Let $(X,\mathcal{A},\mu)$ be a measure space with $\mu(X)< \infty$. $\;$Suppose $\phi: X \longrightarrow [0,\infty) $ is measurable. Let $\epsilon>0$.
Prove that there exists a measurable subset $A$ of $X$ such that $\mu(X\setminus A) < \epsilon$ and $\phi$ is bounded on $A$.
Hint: for $n\in \mathbb{N}$, define $A_n = \{x \in X \;| \; \phi(x) < n \}$. I don't know how to use this hint to prove the claim.
Any help would be appreciated.
Since $A_n\subseteq A_{n+1}$ and $\bigcup_{n\ge 1}A_n=X$, $$ \mu(X)=\lim_{n\to\infty}\mu(A_n), $$ which means that $\mu(X\setminus A_n)=\mu(X)-\mu(A_n)<\epsilon$ for all $n$ large enough.