Let $\{f_n\}_{n=1}^{\infty}$ be a sequence of non-negative continuous functions defined on $[0, 1]$. Assume that $f_n(x) \to f(x$) for every $x \in [0, 1]$. Then if $f_n(x) \leq f(x)$ for every $x \in [0, 1]$, we must have $$\lim_{n \to \infty}\int_0^1 f_n(x) dx = \int_0^1 f(x) dx$$
I know that if $f_n \to f$ uniformly, then above result is true? Is the sequence $\{f_n\}$ converges uniformly or it can be proved without using the uniform convergence?