Let $\mathcal{H}$ be a Hilbert space equipped with inner product $(\cdot,\cdot)$. Suppose that a sequence $u_n\rightarrow u$ weakly in $(\mathcal{H},(\cdot,\cdot))$, I want to prove that $$(u,u)\le \underset{n\rightarrow\infty}{\text{liminf}}(u_n,u_n)$$which is very useful in proving the existence of solutions to certain variational problems.
However I cannot give a direct proof myself. Actually I am not even sure if it is correct. Please help by giving either a proof or a counterexample to the statement above.
