How can I prove that Xn converges to 0 in probability? 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 $\infty$:


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