Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $$ \sum_{n = 1}^\infty \frac{|a_n - b_n|}{2^n}$$

converges.

share|improve this question
    
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
add comment

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?

share|improve this answer
    
it is converges and former series also converges by the comparasion test. Thank you so much –  Eda Yıldız Oct 13 '13 at 8:00
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.