How can I prove that Xn converges to 0 in probability?

Question

Let $X_n\sim U[-1/n,1/n]$. Since for convergence in probability for every $\epsilon>0$,

$$ \lim_{n\to\infty} P(|X_n - X|\ge \epsilon) = 0 $$

Hence, $P(|X_n-0|\ge \epsilon)=1-P(|X_n|<\epsilon)=1-\epsilon n$. But the limit of this function is $\infty$:

Answer 1

$P(|X_n| > \epsilon) = 0$ as soon as $\epsilon > \frac 1n$. Check your computation again.