Very challenging measure theory problem with $L^p$ norms and lower bound for probability

Question

In a probability space X, we are given $f_n$ and $a>0$ so that $1\le\|f_n\|_2\le {1 \over a} \|f_n\|_1$ for every $n\ge 1$. (The $L^2$ and $L^1$ norms respectively)

Prove that $\:P(|f_n(x)|\ge a/2\:\text{ for infinitely many }n)\ge a^2 /4$