Check almost sure convergence

Question

Let random variable $X:U(0,1)$. Check almost sure convergence of sequence of random variables $X_n$ if $X_n=X+I\{0<X<\frac{1}{n^{4/5}}\}$.

Answer 1

Note that $$X_n-X=I\{0<X<n^{-4/5}\}\to 0$$ a.s. when $n\to \infty$. In fact, $$ \{0<X<n^{-4/5}\} \downarrow \emptyset$$ when $n\to \infty$.