To find a relationship between $P($A_n $ i .o)$ and almost convergence we generally use the Borel Cantelli lemma.
But let say $A_n$ = ($X_n \neq $ $Y_n$) where $Y_i= X_i I(|X_i|\leq i)$.
It is given that p( $X_n \neq $ $Y_n $ $i.o$)=$0$.
This implies p( $X_n = $ $Y_n $ $i.o$)=$1$.
My question is this means that $X_n$ - $Y_n$ $\overset{a.s.}{\rightarrow} 0 $ ?
Without proving anything , i think this is correct based on my intuition .
Because p( $X_n = $ $Y_n $ $i.o$)=$1$ means no matter how large the n is , $X_n$ equals to $Y_n$. So is this enough to say that $X_n$ - $Y_n$ $\overset{a.s.}{\rightarrow} 0 $ in this example ?