Measure, Integration & Real Analysis, 2021a

Cauchy Sequences and Completeness
Section 6A Metric Spaces 151
The next definition is useful for showing (in some metric spaces) that a sequence has
a limit, even when we do not have a good candidate for that limit.
6.12 Definition Cauchy sequence
A sequence f 1 , f 2 ,...in a metric space (V, d) is called a Cauchy sequence if for
every ε > 0, there exists n ∈ Z + such that d( f j , f k ) < ε for all integers j ≥ n
and k ≥ n.
6.13 every convergent sequence is a Cauchy sequence
Every convergent sequence in a metric space is a Cauchy sequence.
Proof Suppose lim k→∞ f k = f in a metric space (V, d). Suppose ε > 0. Then
there exists n ∈ Z + such that d( f k , f ) <
2 ε for all k ≥ n. Ifj, k ∈ Z+ are such that
j ≥ n and k ≥ n, then
d( f j , f k ) ≤ d( f j , f )+d( f , f k ) < ε 2 + ε 2 = ε.
Thus f 1 , f 2 ,...is a Cauchy sequence, completing the proof.
Metric spaces that satisfy the converse of the result above have a special name.
6.14 Definition complete metric space
A metric space V is called complete if every Cauchy sequence in V converges to
some element of V.
6.15 Example
• All five of the metric spaces in Example 6.2 are complete, as you should verify.
• The metric space Q, with metric defined by d(x, y) =|x − y|, is not complete.
To see this, for k ∈ Z + let
x k = 1
10 1! + 1
1
+ ···+
102! 10 k! .
If j < k, then
|x k − x j | =
1
1
+ ···+
10
(j+1)! 10 k! <
2
10 (j+1)! .
Thus x 1 , x 2 ,...is a Cauchy sequence in Q. However, x 1 , x 2 ,...does not converge
to an element of Q because the limit of this sequence would have a decimal
expansion 0.110001000000000000000001 . . . that is neither a terminating decimal
nor a repeating decimal. Thus Q is not a complete metric space.
Measure, Integration & Real Analysis, by Sheldon Axler