# Inequality involving the measure of $\displaystyle\limsup_{n\to\infty}(\{x\in{}X:|f_n(x)|\geq{}a/2\})$

Let $$(X,M,\mu)$$ be a measure space with $$\mu(X)=1$$ and $$f_n\in\mathcal{L}^2(X)$$ such that there exists an $$a>0: 1\leq\left\lVert{}f_n\right\rVert_2\leq{}\frac{1}{a}\left\lVert{}f_n\right\rVert_1$$ for all $$n\geq{}1$$. Show that $$\mu(\{x\in{}X:|f_n(x)|\geq{}a/2 \text{ for infinite values of } n\})\geq{}a^2/4$$.

Observations: The condition becomes $$1\leq{}\int_X(f_n)^2d\mu\leq{}(\frac{1}{a}\int_X|f_n|d\mu)^2$$ and setting $$E_n=\{x\in{}X:|f_n(x)|\geq{}a/2\}$$ we observe that we need to show that $$E=\{x\in{}X:|f_n(x)|\geq{}a/2 \text{ for infinite values of } n\} =\displaystyle\limsup_{n\to\infty}E_n$$

Obviously $$a\leq{}1$$ since $$\int_X(f_n)^2d\mu\geq{}(\int_X|f_n|d\mu)^2$$ by Jensen's inequality.

Any ideas about how to proceed?