Proving $X=Y$ almost surely

Question

Given the three conditions: $X_n \overset{a.s.}{\longrightarrow} Y_n$, $X_n \overset{a.s.}{\longrightarrow} X$, $Y_n \overset{L^1}{\longrightarrow} Y$, where $X_n, Y_n, X, Y$ are random variables,

Can we conclude that $X=Y$ almost surely? I think we can conclude this, but I don't have mathematically rigorous proof for this. If the conclusion can't be made, what is the counter example to this?

Thank you,

Answer 1

$X_n \to Y_n$ as is meaningless. I will interpret this as $X_n -Y_n \to 0$ a.s.

$Y_n \to Y$ in $L^{1}$ implies that there is a subsequence $(Y_{n_k})$ which converges to $Y$ a.s. Now $X =\lim X_{n_k}=\lim (X_{n_k}-Y_{n_k}) +Y_{n_k}=Y$ a.s.