How to prove $X_n$ converges to $0$ in probability measure

Question

I'm new to measure theory. I'm pretty struggle with the following question. Really need help to formally prove the question.

$$U\sim \mathrm{Uniform}(0,1),$$ $X_n=\sqrt{n}I_{(0,1/n)}(U)$, how to prove $X_n \rightarrow 0$ in probability measure?

Answer 1

You can also prove that $(X_n)_{n \in \mathbb{N}}$ converges in $L^1(\mathbb{P})$ (in-mean convergence : $\displaystyle\lim_{n \to \infty} \mathbb{E}(|X_n - 0|) \to 0$). Then, you use the fact that in-mean convergence implies probability convergence.