Prove Cauchy sequence & find bounds on limit

*melinda*

Here's the problem statement:

Prove that $x_1,x_2,x_3,...$ is a Cauchy sequence if it has the property that $|x_k-x_{k-1}|<10^{-k}$ for all $k=2,3,4,...$. If $x_1=2$, what are the bounds on the limit of the sequence?

Someone suggested that I use the triangle inequality as follows:

let $n=m+l$
$$|a_n-a_m|=|a_{m+l}-a_m|$$
$$|a_{m+l}-a_m|\leq |a_{m+l}-a_{m+l-1}|+|a_{m+l-1}-a_{m+l-2}|+...+|a_{m+1}-a_m|$$

Now by hypothesis, $|a_k-a_{k-1}|<10^{-k}$, so

$$|a_{m+l}-a_m|<10^{-(m+l)}+10^{-(m+l-1)}+...+10^{-(m+1)}$$.

It looks like we have an $\epsilon$ such that $|a_n-a_m|<\epsilon$. Before we get to the bounds on the limit, is that correct? Is anything missing?

Related Introductory Physics Homework Help News on Phys.org

Tom Mattson

Staff Emeritus
Gold Member
You might take it a little further:

$$|a_{m+l}-a_m|<\sum_{i=0}^l 10^{m+l-i}$$

$$|a_n-a_m|<\sum_{i=0}^{n-m} 10^{n-i}$$

I'll let a real mathematician help you the rest of the way.

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving