Limsup and in probability convergence

Question

Suppose I have two sequences $X_{n}$ and $Y_{n}$. We know that $$\limsup_{n\rightarrow\infty}X_{n}=\infty\quad a.s.\text{ and }|X_{n}-Y_{n}|\overset{p}{\rightarrow}0\text{ as }n\rightarrow\infty.$$ Then can we conclude that $$ \limsup_{n\rightarrow\infty}Y_{n}=\infty\text{ a.s.} $$

Answer 1

This is not true. Let $(X_n)$ be independent with $P(X_n=n)=\frac 1 n$ and $P(X_n=-n)=1-\frac 1 n$. Let $Y_n=-n$ when $X_n=-n$ and $0$ when $X_n=n$. Since $\sum P(X_n=n)=\sum \frac 1n =\infty$ it follows by Borel Cantelli Lemma that $P (\lim \sup X_n=\infty)=1$. Since $Y_n\leq 0$ it follows that $P (\lim \sup Y_n=\infty)=0$. Note that $P(|X_n-Y_n| >\epsilon) \leq P(X_n \neq Y_n)=\frac 1 n \to 0$.