# Cauchy product of series

1. Jun 12, 2009

### Bleys

I don't understand a small part in the proof that two absolutely convergent series have absolutely convergent cauchy product.
Instead of writing the whole thing, I'll write the essentials and the step I'm having trouble with.

$$\sum_{r=1}^{\infty}a_{r}$$ and $$\sum_{r=1}^{\infty}b_{r}$$ are positive term series that are absolutely convergent. Denote their partial sums as $$s_{n} , t_{n}$$ respectively. Let $$w_{n}=s_{n}t_{n}$$ and $$u_{n}=\sum_{r=1}^{\n}c_{r}$$ where $$c_{n}$$ is the Cauchy product of $$a_{n}$$ and $$b_{n}$$

Then $$w_{\lfloor n/2\rfloor}\leq u_{n}\leq w_{n}$$. This is the step I don't understand. I can see why it would be smaller than $$w_{n}$$, since it's a sum containing $$u_{n}$$, but I don't see why it would be greater than $$w_{\lfloor n/2\rfloor}$$.

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