Cauchy Sequence

1. Mar 4, 2008

rbzima

I'm basically trying to show that if (an) and (bn) are Cauchy sequences, then (cn) = |an - bn| is also a Cauchy sequence.

I know that the triangle inequality is going to be used at one point or another, but I suppose I'm a little confused because:

(an) is Cauchy implies |an - am| < e
(bn) is Cauchy implies |bn - bm| < e

I think at some point my e's are going to be changed to e/2, which is totally legitimate because e is arbitrary anyway.

2. Mar 4, 2008

mathman

The key to this is to show that ||X|-|Y||<=|X-Y|

Then ||an-bn|-|am-bm||<=|(an-am)-(bn-bm)|<=|an-am|+|bn-bm|