[SOLVED] Another lim inf/lim sup proof

**Pinkk** · February 24th 2010, 05:18 PM

Let $(s_{n})$ be a sequence of nonnegative numbers, and for each n define $a_{n} = \frac{1}{n}(s_{1} +s_{2}+...+ s_{n})$ . Show that $\lim \,inf \,s_{n} \le \lim \,inf \,a_{n} \le \lim \,sup \,a_{n} \le \lim \,sup \,s_{n}$ . Also show that if $\lim s_{n}$ exists, then $\lim a_{n}$ exists and $\lim a_{n} = \lim s_{n}$ .

For this one I am completely stuck. Any help would be appreciated.

**Drexel28** · February 24th 2010, 05:53 PM

Originally Posted by Pinkk

Let $(s_{n})$ be a sequence of nonnegative numbers, and for each n define $a_{n} = \frac{1}{n}(s_{1} +s_{2}+...+ s_{n})$ . Show that $\lim \,inf \,s_{n} \le \lim \,inf \,a_{n} \le \lim \,sup \,a_{n} \le \lim \,sup \,s_{n}$ . Also show that if $\lim s_{n}$ exists, then $\lim a_{n}$ exists and $\lim a_{n} = \lim s_{n}$ .

For this one I am completely stuck. Any help would be appreciated.

What is the definition of $\limsup$ ?

**Pinkk** · February 24th 2010, 06:13 PM

$lim sup = \lim_{N\to \infty}\, sup\{s_{n} : n > N\}$

**Pinkk** · February 24th 2010, 09:01 PM

Any suggestions?

**Pinkk** · February 25th 2010, 06:45 PM

Apparently, for part of the proof I need to show that $M > N$ implies $sup\{a_{n} : n > M\} \le \frac{1}{M}(s_{1}+s_{2}+...+s_{N}) + sup\{s_{n} : n > N\}$ (and I don't even how to show that or why that's true...

**Pinkk** · February 27th 2010, 04:00 PM

Sorry for all the consecutive posts, but I am still absolutely stuck on this.

Thread: [SOLVED] Another lim inf/lim sup proof

LinkBack

Thread Tools

Display

[SOLVED] Another lim inf/lim sup proof

Similar Math Help Forum Discussions

[SOLVED] direct proof and proof by contradiction

[SOLVED] Proof GCD and LCM

[SOLVED] Log proof

[SOLVED] proof

[SOLVED] Need help on this proof

Search Tags