Does there exist a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and a sequence of real random variables $\{X_n\}_{n\in \mathbb{N}}$ converging in probability, but such that $\mathbb{P}(\{\omega\in \Omega \ |\ \sup_n |X_n| = + \infty \}) > 0 $ ?
I managed to prove that if $X_n \to X$ almost surely then the set $\{\omega\in \Omega \ |\ \sup_n |X_n| = + \infty \}$ is negligible, but I found difficulties in finding a counterexample when $X_n \rightarrow X$ in probability.