# How to write a mathematically rigorous definition of completeness

1. Oct 10, 2009

### royzizzle

I know that the definition of completeness is that a set contains the limits of rational numbers.

and I know the definition of convergence is that for all e>0 there exists N such that for n>=N |xn - x| < e where x is the limit of the sequence.

how to combine the two?

2. Oct 10, 2009

### neb5588

I know the definition in terms of metric spaces, so maybe this is what youre looking for....

A metric space (E,d) is called complete if every Cauchy sequence in E converges in E.

of course the definition for cauchy sequence is a sequence given any e>0
there is a positive integer N such that d(pm,pn) < e whenever n,m >N

3. Oct 10, 2009

### HallsofIvy

There are, in fact, six properties of the real numbers that are equivalent to "completeness".

1. The least upper bound property (every non-empty set having an upper bound has a least upper bound) and its "twin" the greatest lower bound property.

2. Monotone convergence (every increasing sequence having an upper bound converges and its "twin" that every decreasing sequence having a lower bound converges.)

3. The Cauchy Criterion (every Cauchy sequence converges)

4. The Bolzano-Weierstrass property (every bounded sequence contains a convergent subsequence.)

5. Every closed and bounded set is compact.

6. The set of all real numbers, with the usual metric, is a connected set.

Given any one of those you can prove the other five.

Last edited by a moderator: Oct 11, 2009