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 !!

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 $\displaystyle |A|-|B|\le |A-B|$.

So $\displaystyle |x_n|\le |x_N|+1$