How do I show that the sequence below is a Cauchy sequence?

Question

If $(N, \|.\|)$ is a normed space and $(x_n)$ a sequence in $N$ such that

$\|x_n - x_{n+1}\| < \frac{1}{2^n}$

Then $(x_n)$ is a Cauchy sequence.

Just a hint on how to prove this will suffice, thank you.

score 3 · Accepted Answer · answered Apr 09 '15 at 13:49

3

Hint. For $n < m$, we have $\def\norm#1{\left\|#1\right\|}$ \begin{align*} \norm{x_n - x_m} &= \norm{\sum_{k=n}^{m-1} x_{k+1} - x_k}\\ &\le \sum_{k=n}^{m-1} \norm{x_{k+1} - x_k} \end{align*} Now use what you are given and what you know aobut the geometric series and its partial sums.

answered Apr 09 '15 at 13:49

martini

84,101

Your sum is < sum 1/2^k k from n to m-1 < sum 1/2^k k from n to infinity =0 and this completes the proof? – bananana Apr 09 '15 at 14:04
Replace $=0$ and the end with $\to 0$ for $n \to \infty$ ... yes. – martini Apr 09 '15 at 14:06
Thank you very much for your help, sir. – bananana Apr 09 '15 at 14:43

How do I show that the sequence below is a Cauchy sequence?

1 Answers1