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"If a sequence a_1 + a_2 + ... converges and if b_1, b_2, ... is a bounded monotonic sequence of numbers, then (a_1)(b_1) + (a_2)(b_2) + ... converges"

Proof:

Let s_n denote the partial sums of [tex] \sum_{v=1}^n a_v [/tex], s the sum, and let [tex] \xi_n = s_n - s [/tex]. Then [tex] \sum_{v=n}^m a_v b_v = \sum_{v=n}^m (\xi_v - \xi_{v-1}) b_v = \sum_{v=n}^m \xi_v(b_v - b_{v+1}) - \xi_{n-1} b_n + \xi_m b_{m+1} [/tex].

For every sufficiently large v, [tex] |\xi_v| < \epsilon [/tex], and

[tex] \sum_{v=n}^m a_v b_v < \epsilon \sum_{v=n}^m |b_v - b_{v+1}| + \epsilon |b_n| + \epsilon |b_{m+1}| < \epsilon |b_n - b_{m+1}| + \epsilon |b_n| + \epsilon |b_{m+1}| [/tex].

This is in turn less than [tex] 4B \epsilon [/tex], where B is a bound for |b_v|, and the series [tex] \sum_{v=1}^{\infty} a_v b_v [/tex] converges

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I understand the proof and everything. I was wondering though, how did the writer of the proof know to rewrite the sum as this: [tex] \sum_{v=n}^m a_v b_v = \sum_{v=n}^m (\xi_v - \xi_{v-1}) b_v = \sum_{v=n}^m \xi_v(b_v - b_{v+1}) - \xi_{n-1} b_n + \xi_m b_{m+1} [/tex] ?

It just seems so random, something that I never would've thought about. If you could, could you please explain the thought processes he went through to realize he had to rewrite the sum in that form?

Thanks