problem with lim of a sequence

Question

i don't know how to prove this lim..i thought i can use Cesaro-Stolz here but i can't..: $$ \lim_{n \to \infty }a_{n}= \infty \\ b_{n}= \frac{1}{n}\sum_{k=1}^{\infty }a_{k} $$ how to prove that : $\lim_{n \to \infty }b_{n}= \infty$

Answer 1

Of course, $$ b_{n}= \frac1n\sum_{k=1}^na_k. $$ For every $C$ there exists some $N$ such that for every $n\gt N$, $a_n\geqslant C$. Hence, for every $n\geqslant N$, $$ b_n=\frac1n\sum_{k=1}^Na_k+\frac1n\sum_{k=N+1}^na_k\geqslant\frac1n\sum_{k=1}^Na_k+\frac{n-N}nC. $$ The first term on the RHS converges to zero when $n\to\infty$, since $N$ is fixed once $C$ is chosen. What does the second term on the RHS converge to? Can you conclude?