Question on dominating convergence theorem

On $[0,1]$ we have sequence of non-negative continuous functions {${f_n}$ } converging pointwise to $f(x)$,and $f_n(x)\leq f(x)~\forall~x\in[0,1].$Is this condition enough to conclude the following$$\lim_{n\to\infty}\int_{0}^{1}{f_n}dx=\int_{0}^{1}{f}dx.$$

I think to use dominated convergence theorem but couldn't prove that $f\in L^1[0,1]$.Alternatively i am thinking to construct an increasing subsequence by using the condition $f_n(x)\leq f(x)$ and then apply monotone convergence theorem, but then to prove the full result i have to prove that left hand side of above result is convergent.