(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Define the Sequence a_{n}= n^{1/2}where n is a natural number.

Show that |a_{n+1}-a_{n}| -> 0 but a_{n}is not a cauchy sequence

2. Relevant equations

3. The attempt at a solution

(Ignore this paragraph)Well, unfortunately I am stuck on the very first part. How exactly do I evaluate the limit as n -> infinity of |(n+1)^(1/2) - n^(1/2)| ? any hint at a trick would be most welcome (unless of course I am not seeing something that is obvious).

As for the rest, I need to show a_{n}is not a Cauchy sequence. The definition of a Cauchy sequence uses two sequences with different subscripts, m and n. In this case, can I take n+1 to be m and keep n as itself?

I think I need to show that the distance between the two sequences, a_{n+1}and a_{n}is not decreasing as n becomes large.

edit: found limit.

Also, re-reading the question I see a flaw in my above statement. I just need to show that for any given m and n, as I vary them independently, they do not meet the cauchy criterion. The problem itself states that looking only at the n+1 termappearsto meet the criterion, but in fact does not.

Confirm or deny?

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# Homework Help: Cauchy Sequence

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