A Cauchy sequence of real numbers is bounded

• Nov 16th 2012, 01:10 AM
matemauch
A Cauchy sequence of real numbers is bounded
Hello,

There is one step that I did not get in the following proof I found.

Let Xn be a sequence of real numbers. If Xn is Cauchy then the sequence is bounded.

Proof:
Given 1 there exist N such that for all n>=N, |xn-xN|<1.
It follows that|xn|<=|xN|+1.
Let b the maximum of |x1|,...,|xN|,|XN|+1. Then b is a bound for the sequence.

I am facing troubles understanding how did you get from |xn-xN|<1 to |xn|<=|xN|+1
Any hint will be very appreciated !!
• Nov 16th 2012, 04:47 AM
Plato
Re: A Cauchy sequence of real numbers is bounded
Quote:

Originally Posted by matemauch
I am facing troubles understanding how did you get from |xn-xN|<1 to |xn|<=|xN|+1
Any hint will be very appreciated !!

It is very simple.
We know that $|A|-|B|\le |A-B|$.
So $|x_n|\le |x_N|+1$