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7 Variational Principles
Let {x n } be a number sequence in the metric space (X, ρ), if for any ε > 0,
there is always the positive integer N = N(ε), when m, n N, there is identically
|x m − x n | < ε or ρ(x m , x n ) < ε, then the {x n } is called a Cauchy sequence in
(X, ρ). Sometimes {x n } is also called the fundamental sequence in (X, ρ).
About the relationship between the convergent sequence and Cauchy sequence
and the property of the Cauchy sequence, there is the following theorem:
Theorem 7.2.1 Let {x n } be a number sequence in the metric space (X, ρ), (1) If
{x n } is the convergent sequence, then {x n } a Cauchy sequence; But the converse
proposition may not true; (2) If a subset sequence {x n k } in the Cauchy sequence {x n }
converges to x, then the number sequence {x n } also converges to x.
Proof (1) Let {x n } converge to x, then for any ε > 0, there is a positive integer
N, such that when m, n > N, there is identically ρ(x m , x) <
ε
2
and ρ(x n , x) <
ε
2
,
according to the triangle inequality, there is
ρ(x m , x n ) ρ(x m , x) + ρ(x n , x) <
ε
2
+
ε
2
= ε
therefore, {x n } is a Cauchy sequence.
Now giving an example to illustrate that the converse proposition may not true.
Let (X, ρ) be the open interval (0,1), the distance between any points x and y is
defined as ρ(x, y) = |x − y|. Consider the number sequence
1
n
in X , when m,
n → ∞, according to the property of the inequality, there is
ρ
1
m
,
1
n
=
1
m
−
1
n
1
m
+
1
n
→ 0
Therefore
1
n
is a Cauchy sequence in (X, ρ). but lim
n→∞
1
n
= 0 /
∈ (X, ρ), This
shows that the number sequence
1
n
has not limit in (X, ρ), namely
1
n
does not
converge in (X, ρ).
(2) Let {x n } be a Cauchy sequence in (X, ρ), its a subsequence {x n k } converges
to x, that is lim
k→∞
x n k = x ∈ (X, ρ). For any ε > 0, since x n k → x, there is a positive
integer N 1 , such that for all n k > N 1 , there are ρ(x n k , x) <
ε
2
; Moreover since {x n } is
a Cauchy sequence, there is a positive integer N 2 , such that for all m, n > N 2 , there
are ρ(x m , x n ) <
ε
2
. Putting N = max(N 1 , N 2 ), then for all n > N and any n k > N,
there is
ρ(x n , x) ρ(x n , x n k ) + ρ(x n k , x) <
ε
2
+
ε
2
= ε
Hence lim
n→∞
x n = x. Quod erat demonstrandum.
If every Cauchy sequence in a metric space (X, ρ) converges to an element in
(X, ρ), the (X, ρ) is called the complete metric space, it is called the complete