You recall the definition of completeness which stated that every nonempty set of real numbers which is bounded above has a least upper bound and that every nonempty set of real numbers which is bounded below has a greatest lower bound and this is a property of the real line known as the completeness axiom. Geometrically, this involved filling in the holes. There is another way of describing completeness in terms of Cauchy sequences which will be discussed soon.

A sequence is Cauchy means the terms are “bunching up to each other” as m,n get large.
Proof: Let ε = 1 in the definition of a Cauchy sequence and let n > n_{1}. Then from the definition,

It follows that for all n > n_{1},

Therefore, for all n,

Proof: Let ε > 0 be given and suppose a_{n} → a. Then from the definition of convergence, there exists n_{ε} such that if n > n_{ε}, it follows that

Therefore, if m,n ≥ n_{ε} + 1, it follows that

showing that, since ε > 0 is arbitrary,
The following theorem is very useful.
Proof: Let ε > 0 be given. There exists N such that if m,n > N, then

Also there exists K such that if k > K, then

Then let k > max
The next definition has to do with sequences which are real numbers.
Definition 4.9.5 The sequence of real numbers,
If someone says a sequence is monotone, it usually means monotone increasing.
There exist different descriptions of completeness. An important result is the following theorem which gives a version of completeness in terms of Cauchy sequences. This is often more convenient to use than the earlier definition in terms of least upper bounds and greatest lower bounds because this version of completeness, although it is equivalent to the completeness axiom for the real line, also makes sense in many situations where Definition 2.10.1 on Page 61 does not make sense, ℂ for example because by Problem 12 on Page 86 there is no way to place an order on ℂ. This is also the case whenever the sequence is of points in multiple dimensions.
It is the concept of completeness and the notion of limits which sets analysis apart from algebra. You will find that every existence theorem in analysis depends on the assumption that some space is complete.
Theorem 4.9.6 Every Cauchy sequence in ℝ converges if and only if every nonempty set of real numbers which is bounded above has a least upper bound and every nonempty set of real numbers which is bounded below has a greatest lower bound.
Proof: First suppose every Cauchy sequence converges and let S be a nonempty set which is bounded above. Let b_{1} be an upper bound. Pick s_{1} ∈ S. If s_{1} = b_{1}, the least upper bound has been found and equals b_{1}. If

Therefore, if m > n

and so

which would prevent
Now suppose the condition about existence of least upper bounds and greatest lower bounds. Let
Theorem 4.9.7 If either of the above conditions for completeness holds, then whenever
Proof: Let a = sup
By Theorem 4.9.6 the following definition of completeness is equivalent to the original definition when both apply.
Definition 4.9.8 Whenever every Cauchy sequence in some set converges, the set is called complete.
Proof: Suppose