Define the Sequence an= n1/2 where n is a natural number.
Show that |an+1-an| -> 0 but an is not a cauchy sequence
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 an 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, an+1 and an 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 term appears to meet the criterion, but in fact does not.
Confirm or deny?