This would probably look like a dumb question (akin to prove $2=1$ type). But I'd still like to know where the flaw lies. The question is regarding convergence in probability and almost sure convergence. We know the following: Suppose events $A_n \to A$, then $P(A_n) \to P(A)$ i.e $\lim P(A_n) = P(\lim A_n) = P(A)$.
Can't the same principle be used to conclude equivalence of both type of convergence?
$P(\lim X_n = X) = \lim P(X_n = X)$, implying equivalence? Thanks!