weak convergence implies boundedness.

I have these in books without proof, mostly as a corollary. I was wondering if I could get a proof.

Suppose $$\lim_{n\to \infty} \int_0^1 f_ng dx = \int_0^1 fg dx$$ for all $g\in L^2(0,1)$, where $f_n, f \in L^2(0,1)$. Then there exists a constant $K$ such that $\|f_n\|_{L^2} \leq K \lt \infty$ for all $n$.

Solutions Collecting From Web of "weak convergence implies boundedness."