If is an infinite sequence of real numbers such that
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exists and is finite, then we have for all and that
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Let denote the partial sums of the x's. Using summation by parts,
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Pick any ε > 0. Now choose N so that is ε-close to s for k > N. This can be done as the sequence converges to s. Then the right hand side is:
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Now, let n go to infinity. The first term goes to s, which cancels with the third term. The second term goes to zero (as the sum is a fixed value). Since the b sequence is increasing, the last term is bounded by .