Show something converge to infinity strongly

Question

Show that if $ Var(Y_n)=1 $ and $ \mu_n\to \infty $ quickly enough so that $ \sum_{n=1}^{\infty}\frac{1}{\mu_n^2} $ is finite, then $ p[Y_n\to \infty]=1. $

I know that I need to use Chebyshev's inequality and the Borel-Cantelli lemma to show that $P[Y_n < a\ \textrm{for infinitely many }$n$]=0 $ for all a, and the fact $\sum_{n=1}^{\infty}\frac{1}{\mu_n^2} $ is finite, must be useful.

Can anyone give a hint on how to start?

Answer 1

With $E_n=\{|Y_n-\mu_n|>\mu_n/2\}$, $\Sigma_nP(E_n)<\Sigma_n4/\mu_n^2<\infty$. So for large $n$ $Y_n>\mu_n/2$ a.s.