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?

 

[quantumdude]

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.