Proving two limits equal to each other?

Question

I have come across this proof that is a bit tricky to me. The value of An is making it pretty hard.

Prove if $\lim\limits_{n\to \infty} S_n = S$, then $\lim\limits_{n\to \infty} A_n = S$ where $A_n = \frac{S_1 + S_2 + \cdots + S_n}{n}$ ?

I was thinking to use $\varepsilon-N$ definition but am pretty stuck.

Does this answer your question? On Cesàro convergence: If $ x_n \to x $ then $ z_n = \frac{x_1 + \dots +x_n}{n} \to x $ — Elliot Yu, Mar 05 '21 at 18:59
As @Elliot Yu pointed out Cesaro convergence is related, Cesaro Summation might also be helpful to prove it — Jan Safronov, Mar 05 '21 at 19:01

score 0 · Answer 1 · answered Mar 05 '21 at 19:06

We can see this problem as an application of $Cesaro$ criterion.

Let $(S_{n})$ be a sequence of real numbers which converges to $S$, define

$${\displaystyle A_{n}:=\sum _{m=1}^{n}S_{m}=S_{1}+\dots +S_{n},\quad b_{n}:=n}$$ then $\displaystyle (b_{n})$ is strictly increasing and diverges to $\displaystyle +\infty $ . Now we compute

$$\displaystyle \lim _{n\to \infty }{\frac {A_{n+1}-A_{n}}{b_{n+1}-b_{n}}}=\lim _{n\to \infty }S_{n+1}=\lim _{n\to \infty }S_{n}=S$$

and so

$$\displaystyle \lim _{n\to \infty }{\frac {S_{1}+\dots +S_{n}}{n}}=\lim _{n\to \infty }S_{n}$$

So in general given any sequence $\displaystyle (S_{n})_{n\geq 1}$ of real numbers, suppose that

$$\displaystyle \lim _{n\to \infty }S_{n}$$ exists (finite or infinite), then $$\displaystyle \lim _{n\to \infty }{\frac {S_{1}+\dots +S_{n}}{n}}=\lim _{n\to \infty }S_{n}=S$$

Proving two limits equal to each other?

1 Answers1