## 3.2. Cauchy Sequences

### 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*.

### 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:

if| a_{j}- a_{k}| | a_{j}- a | + | a - a_{k}| < 2 / 2 =

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

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

Since the sequence is bounded (by part one of the theorem), say by a constant= {xS: x < aR_{j}for all j except for finitely many}

*M*, we know that every term in the sequence is bigger than

*-M*. Therefore

*-M*is contained in

*. Also, every term of the sequence is smaller than*

**S***M*, so that

*is bounded by*

**S***M*. Hence,

*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*

**S**We will now show that thisa = sup()S

*a*is indeed the limit of the sequence. Take any

*> 0*, and choose an integer

*N > 0*such that

if| a_{j}- a_{k}| < / 2

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

if| a_{j}- a_{N + 1}| < / 2

*j > N*, or equivalently

Hence we have:- / 2 < a_{j}- a_{N + 1}< / 2

fora_{j}> a_{N + 1}- / 2

*j > N*. Thus,

*a*is in the set

_{N + 1}- / 2*, and we have that*

**S**It also follows thata a_{N + 1}- / 2

fora_{j}< a_{N + 1}+ / 2

*j > N*. Thus,

*a*is not in the set

_{N + 1}+ / 2*, and therefore*

**S**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_{N + 1}| < / 2

for any| a - a_{j}| < |a - a_{N + 1}| + | a_{N + 1}- a_{j}| < 2 / 2 =

*j > N*.

Contributed to this page:*
Lakshmi Natarajan
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