How to show that this seq. of r.v.-s $X_n=n 1_{\{[0, \frac{1}{n}\}}$ converge to $0$ a.s.?
The definition of almost sure convergence of $X_n$ to $0$ is that $$ P(\{ \omega:\lim_{n}X_n(\omega)=0\})=1. $$ For $\omega\ne 0$, $\lim_nX_n(\omega)=0$, and $P(\{0\})=0$ (assuming $P$ is Lebesgue measure of course).