I have been trying to disprove that if I have a sequence of random variables $X_n$, that $X_n \to a$, where $a$ is a constant, in distribution doesn't imply $X_n \to a$ almost surely. One example I came up with was where $X_n \sim N(0, n)$. $X_n$ converges in distribution to $1/2$, but I am not sure how to show almost convergence fails here. Does anyone have a hint or know if this is a valid example? Thanks.
For exemple , take the sequence where $X_n$ are independent bernoulli variables such that: $$P(X_n = 1 ) = 1/n$$ So , Borel-Cantelli tells you that the event $(X_n =1 )$ will happen infintely many times but $P(X_n=0)=1-1/n$ converges to 1 , so the sequence converges in distribution to 0