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Denote by $E$ the set of all real sequences $\{a_n\}$ such that $|a_n| ≤ 1$ for every $n\in\mathbb{N}$.

Let $\{a_n\}$, $\{b_n\} ∈ E$. Prove that the series $$ \sum_{n = 1}^\infty \frac{|a_n - b_n|}{2^n}$$


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This (and many similar questions) are worked out in most textbooks on mathematical analysis. For example, you can try W. Rudin, Principles of Mathematical Analysis, or Apostol's book on analysis. –  Doldrums Oct 13 '13 at 18:22
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1 Answer

First note that $|a_n - b_n| \leq 2$ for all $n$ by the triangle inequality. Then we can bound the series by $\sum_{n=1}^\infty \frac{|a_n - b_n|}{2^n} \leq 2 \sum_{n=1}^\infty \frac{1}{2^n}$. What do you know about the latter series?

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it is converges and former series also converges by the comparasion test. Thank you so much –  Eda Yıldız Oct 13 '13 at 8:00
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