Sequence of empirical means converges to the empirical mean of the limits?

Question

Let $S_n=1/n\sum_{i=1}^n X_{i,n}$ , where $X_{i,n}$ are random variables such that for all $i=1,\cdots \infty$ we have $X_{i,n}\underset{n\rightarrow\infty}{\longrightarrow}0$ almost surely. Doest $S_n \underset{n\rightarrow\infty}{\longrightarrow}0$ almost surely?

Answer 1

Not true even for constant r.v.'s Take $X_{i,n}=\frac i n$, for example. $ S_n \to \frac 1 2$ in this case.