$X_n$ does not converge in probability

Question

Let $X$ be a random variable with uniform distribution at $(0,1)$ and $X_n$ with uniform distribution at $(0, 1+1/n)$.

a) $X_n$ converges in distribution to $X$.

b) Prove that $X_n$ does not converges in probability to $0$.

I was able to did the first exercise, but i'm stuck with b). I think I have to prove that $\lim_{n \to \infty} P[|X_n|<\epsilon]$ does not converges to $1$. I would be thankful with any help.

Answer 1

Hint: for any $\epsilon$ satisfying $0 < \epsilon < 1$, $$P(|X_n| < \epsilon) = P(0 \leq X_n < \epsilon) = \int_{0}^{\epsilon}\dfrac{1}{1+1/n}\text{ d}x = \dfrac{\epsilon}{1+1/n}\text{.}$$ Now take $n \to \infty$.