Uniqueness of the limit on random variable

A quick proof: if $X_n \to X$ a.s. and $E|X_n-Y|^{2} \to 0$ then Fatou's Lemma gives $E|X-Y|^{2} \leq \lim \inf E|Y-X_n|^{2}=0$ so $X=Y$ a.s.