How can I show that $b_n \rightarrow a$?

Question

Knowing that $a_n \rightarrow a$ and $b_n=\frac{a_1+a_2+...+a_n}{n}$. How can I show that $b_n \rightarrow a$?

Do you want an exact or intuitive proof? – Ragnar Feb 07 '14 at 18:58 — Ragnar, Feb 07 '14 at 18:58
@Ragnar I want an exact proof. – Mary Star Feb 07 '14 at 19:00 — Mary Star, Feb 07 '14 at 19:00
Here's the first duplicate I found. – David Mitra Feb 07 '14 at 19:10 — David Mitra, Feb 07 '14 at 19:10

score 1 · Answer 1 · answered Feb 07 '14 at 19:24

Easily $$\sup_{1\le i\le n}a_i \ge b_n = \dfrac{a_1+a_2+\ldots+a_n}{n}\ge\inf_{1\le i\le n}a_i$$ As $\displaystyle\lim_{n\to\infty} a_n=a\implies\limsup_{n\to\infty}a_n=\liminf_{n\to\infty}a_n=a$.

Thus by Sandwich theorem we have our desired result.

How can I show that $b_n \rightarrow a$?

1 Answers1