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absolutely convergent sequences

Source: Iran PPCE 2012-Analysis exam-P6

February 14, 2012
limitreal analysisreal analysis unsolved

Problem Statement

Suppose that aija_{ij} are real numbers in such a way that for each ii, the series ∑j=1∞aij\sum_{j=1}^{\infty}a_{ij} is absolutely convergent. In fact we have a series of absolutely convergent serieses. Also we know that for each bounded sequence {bj}j\{b_j\}_j we have lim⁡i→∞∑j=1∞aijbj=0\lim_{i\to \infty} \sum_{j=1}^{\infty}a_{ij}b_j=0. Prove that
lim⁡i→∞∑j=1∞∣aij∣=0\lim_{i\to \infty}\sum_{j=1}^{\infty}|a_{ij}|=0.
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