# Prove that $\lim_n \frac{a_1 + \cdots + a_n}{n}=L$ [closed]

Let $a_n$ be a bounded sequence such that $\left|a_n\right| < M$ for all $n$. Let $T_k$ be an integer sequence such that $\lim_{k\to\infty} T_k = \infty$ but $\lim_k \frac{T_k}{T_{k+1}} = 1$. Show that

$$\lim_k \frac{a_1+a_2+\cdots a_{T_k}}{T_k}=L\implies \lim_n \frac{a_1 + \cdots + a_n}{n}=L$$

This reminds me of Cesaro mean, but not quite, since $a_n$ are not necessarily convergent. I think I need to use $\lim \frac{T_k}{T_{k+1}} = 1$, but not sure how.

## closed as off-topic by Did, Claude Leibovici, Juniven, Namaste, Mostafa AyazFeb 5 '18 at 19:10

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Did, Claude Leibovici, Juniven, Namaste, Mostafa Ayaz
If this question can be reworded to fit the rules in the help center, please edit the question.

Yes, this is certainly false without the assumption $\lim T_k/T_{k+1}=1$.
Hint: Let's write $$\sigma_n=\frac{a_1+\dots+a_n}{n}$$and
$$t_k=\sigma_{T_k}.$$
Suppose $T_k\le n<T_{k+1}$. Then $$\sigma_n=\frac{T_k}{n}t_k+\frac{a_{{}_{T_k+1}}+\dots+a_{n}}{n}=I+II.$$
Note that $$|II|\le\frac{n-T_{k}}{n}M\le\frac{T_{k+1}-T_k}{T_k}M,$$ so $$|\sigma_n-t_k|\le\left(\frac{T_k}n-1\right)|t_k|+\frac{T_{k+1}-T_k}{T_k}M.$$
• What is $s$ though? Did you mean $\sigma$? – Yuki Kawabata Feb 10 '18 at 0:49