You can see that as n -> infinity, the probability of Y = 0 approaches 1. This is because as n gets bigger, 1/(n^2) gets smaller and smaller.
Almost sure convergence is defined as
So it can be seen that P(lim n->inf (Yn) = 0) = 1 because lim(1/(n^2)) = 0. I'm not sure how much more you have to prove there, I think all you should have to do is plug in what Yn is to that equation and show what happens as n->inf.
you have something stronger here
you have complete convergence
Complete Convergence and the Law of Large Numbers
which via Borel-Cantelli implies a.s. convergence
If you prove that, almost-surely, for all large enough, then of course this implies that almost-surely .
Let . Prove that Borel-Cantelli lemma applies to the sequence of events . This will imply what I said above.