- #1

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Instead of writing the whole thing, I'll write the essentials and the step I'm having trouble with.

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

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