Counter example of Lebesgue dominated convergence theorem if $f_n$ is not bounded by a integrable function [duplicate]

Let $f$ and $f_n$ be nonnegative measurable function on $[0,1]$ such that $f_n\to f$ pointwise, and we may assume that $f_n\leq f$ for all $n$ in $\mathbb{N}$.

How to find a counterexample to show that $\int f_n d\mu\to \int f d\mu$ is WRONG in this condition, because we do not have an integrable function $g$ that dominates $f_n$.

If this proposition is true, how to prove it, although I think this proposition is wrong.