I read this article about almost sure inequality and the supremum from Almost sure inequality and the supremum
Let $X_n$ be a sequence of random variables, if $\left|X_n \right| \leq Y$ almost surely for all $n$, then $\sup \left| X_n \right| \leq Y$ almost surely.
Is it true that if $X_n$ converges to $X$ in probability, then $|X| \leq \sup_{k \geq 1} \left| X_{n_k} \right| \leq \sup \left|X_n \right| \leq Y$ where $X_{n_k}$ is a subsequence?