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|<1to|xn|<=|xN|+1

Any hint will be very appreciated !!