2.2 Cauchy Sequences
Of course it does not go the other way. For example, you could let xn =
and it has a
convergent subsequence but fails to converge. Here d
and the metric space is
However, there is a kind of sequence for which it does go the other way. This is called a
is called a Cauchy sequence if for every ε >
0 there exists N
such that if m,n ≥ N, then
Now the major theorem about this is the following.
Theorem 2.2.2 Let
be a Cauchy sequence. Then it converges if and only
if any subsequence converges.
This was just done above.
= Suppose now that
is a Cauchy sequence
. Then there exists N1
such that if k > N1,
the definition of what it means to be Cauchy, there exists N2
such that if m,n ≥ N2,
2. Let N ≥
. Then if
k ≥ N,
then nk ≥ N
It follows from the definition that limk→∞xk = x. ■
Definition 2.2.3 A metric space is said to be complete if every Cauchy
Another nice thing to note is this.
Proposition 2.2.4 If
is a sequence and if p is a limit point of the set S
then there is a subsequence
Proof: By Theorem 2.1.7, there exists a sequence of distinct points of S denoted as
such that none of them equal
. Thus B
infinitely many different points of the set
, this for every r.
Let xn1 ∈ B
is the first index such that xn1 ∈ B
chosen, the ni
increasing and let 1 > δ1 > δ2 >
where xni ∈ B
Let xnk+1 ∈ B
is the first index such that xnk+1
is contained B
Another useful result is the following.
Lemma 2.2.5 Suppose xn → x and yn → y. Then d
Proof: Consider the following.
and the right side converges to 0 as n →∞. ■