Fatou lemma and weak convergence in Hilbert

Question

In a Hilbert space $H$ a sequence $(x_n)_{n\geq0}$ is said to converge weakly to $x$ if $\forall y\in H:\langle y,x_n\rangle\rightarrow\langle y,x\rangle$, the case in which we can easily deduce an inequality: $$|x|\leq\lim_{n\rightarrow\infty}\inf_{m\geq n}|x_m|$$ by Schwartz inequality. Is there any connection between this inequality and Fatou's lemma?