Theorem 3.2.2: Completeness Theorem in R

Let be a Cauchy sequence of real numbers. Then the sequence is bounded.
Let be a sequence of real numbers. The sequence is Cauchy if and only if it converges to some limit a.
Context

Proof:

The proof of the first statement follows closely the proof of the corresponding result for convergent sequences. Can you do it ?

To prove the second, more important statement, we have to prove two parts:

First, assume that the sequence converges to some limit a. Take any > 0. There exists an integer N such that if j > N then | a_j - a | < /2. Hence:

| a_j - a_k | | a_j - a | + | a - a_k| < 2 / 2 =

if j, k > N. Thus, the sequence is Cauchy.

Second, assume that the sequence is Cauchy (this direction is much harder). Define the set

S = {x R: x < a_j for all j except for finitely many}

Since the sequence is bounded (by part one of the theorem), say by a constant M, we know that every term in the sequence is bigger than -M. Therefore -M is contained in S. Also, every term of the sequence is smaller than M, so that S is bounded by M. Hence, S is a non-empty, bounded subset of the real numbers, and by the least upper bound property it has a well-defined, unique least upper bound. Let

a = sup(S)

We will now show that this a is indeed the limit of the sequence. Take any > 0 , and choose an integer N > 0 such that

| a_j - a_k | < / 2

if j, k > N. In particular, we have:

| a_j - a_{N + 1} | < / 2

if j > N, or equivalently

- / 2 < a_j - a_{N + 1} < / 2

Hence we have:

a_j > a_{N + 1} - / 2

for j > N. Thus, a_{N + 1} - / 2 is in the set S, and we have that

a a_{N + 1} - / 2

It also follows that

a_j < a_{N + 1} < / 2

for j > N. Thus, a_{N + 1} < / 2 is not in the set S, and therefore

a a_{N + 1} < / 2

But now, combining the last several line, we have that:

|a - a_{N + 1} | < / 2

and together with the above that results in the following:

| a - a_j | < |a - a_{N + 1} | + | a_{N + 1} - a_j | < 2 / 2 =

for any j > N.

Interactive Real Analysis, ver. 1.9.5
(c) 1994-2007, Bert G. Wachsmuth
Page last modified: Mar 28, 2007