Kronecker's lemma/Proof
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For the statement of the lemma, see Kronecker's lemma.
Let Sk denote the partial sums of the x's. Using summation by parts,
Pick any ε > 0. Now choose N so that Sk is ε-close to s for k > N. This can be done as the sequence Sk converges to s. Then the right hand side is:
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 .