1. ## Analysis: alternating series

Let $\displaystyle (a_n)_{n\in\mathbb{N}}$ be a non-increasing non-negative sequence.

Prove that $\displaystyle s_n= \sum_{i=1}^{n}{(-1)^{i+1}a_i}$ is bounded and that $\displaystyle \limsup{s_n}-\liminf{s_n}=\lim{a_n}$

I would appreciate any help.

2. Originally Posted by JoachimAgrell
Let $\displaystyle (a_n)_{n\in\mathbb{N}}$ be a non-increasing non-negative sequence.

Prove that $\displaystyle s_n= \sum_{i=1}^{n}{(-1)^{n+1}a_i}$ is bounded and that $\displaystyle \limsup{s_n}-\liminf{s_n}=\lim{a_n}$

I would appreciate any help.
Maybe I'm reading this wrong but what about $\displaystyle a_i=1$, it's clearly non-increasing and non-negative but $\displaystyle s_n=(-1)^{n+1}n$ which is clearly not bounded

3. Oops. I meant (-1)^{i+1} in that series.

4. Originally Posted by JoachimAgrell
Let $\displaystyle (a_n)_{n\in\mathbb{N}}$ be a non-increasing non-negative sequence.

Prove that $\displaystyle s_n= \sum_{i=1}^{n}{(-1)^{i+1}a_i}$ is bounded and that $\displaystyle \limsup{s_n}-\liminf{s_n}=\lim{a_n}$

I would appreciate any help.
Okay here's my attempt:

By induction we prove that $\displaystyle s_n\leq a_1$ and thus we get that $\displaystyle s_n$ is bounded because it's non negative. Now it's clear that $\displaystyle s_{2n-1} \geq s_{2n}$ for all $\displaystyle n$ and since each of these are monotonic we get that $\displaystyle s_n$ has at most two acc. points and so $\displaystyle s_{2n-1} \rightarrow \limsup s_n$ and $\displaystyle s_{2n} \rightarrow \liminf s_n$ but $\displaystyle s_{2n-1} -s_{2n}= a_{2n} \rightarrow \lim a_n$