Cauchy product of series

**Bleys** · Jun12-09, 03:38 PM

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.

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

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