Let $b_n=\frac1n\sum_{i=1}^n\xi_{ni},$ if $\lim_{n\to\infty}\xi_{ni}=0$, do we have $\lim_{n\to\infty}b_n=0?$

Question

Let $$b_n=\frac1n\sum_{i=1}^n\xi_{ni}.$$ If for each $i=1,\cdots,n$, we have $$\lim_{n\to\infty}\xi_{ni}=0.$$ Can we claim that $$\lim_{n\to\infty}b_n=0?$$

Answer 1

Since for each $i=1,\cdots,n$, we have $\lim_{n\to\infty}\xi_{ni}=0$, then for any $\varepsilon>0$ and $i=1,\cdots,n$, there exists $N_i$ s.t. for $n>N_i$, we have $$|\xi_{ni}|<\varepsilon.$$ Let $N=\max_{1\le i\le n}{N_i}$, then for $n>N$ and all $i=1,\cdots,n$, we have $$|\xi_{ni}|<\varepsilon.$$ Furthermore, we have $$|b_n|\le\frac1n\sum_{i=1}^n|\xi_{ni}|<\frac1n\sum_{i=1}^n\varepsilon=\varepsilon.$$

Answer 2

No (with $\xi_{ni}=i^2/n$, say, you have $\displaystyle\lim_{n\to\infty}b_n=\infty$).