COMPLETE METRIC SPACE #complete #metric $space

December 02, 2025

COMPLETE METRIC SPACE :

A metric space X with the metric d is called a complete metric space

if every cauchy sequence in x converges in X. ∴

For Example : 1. consider the metric space (0,1) and {1/n} is a cauchy sequence in (0,1).

We know that the sequence {1/n} converges to 0.

since this limit point 0 does not lies in (0,1) , (0,1) is not a complete metric space.

Instead of (0,1) if we consider the metric space [0,1], the above mensioned limit

point 0 lies in [0,1] and hence [0,1] is a complete metric space.

2. The real line is a complete metric space.

3. The complex plane is also complete which is explained in the following argument .

Let { z_n } , where z_n = a_n + b_n , be a Cauchy sequence of complex numbers. Then { a_n } and { b_n } are themselves Cauchy sequences of real numbers.

Since the real line is complete and { a_n } and { b_n } are Cauchy sequences in the real line R, the sequences { a_n } and { b_n } and convergent to the real numbers a and b (Say ) i.e. a_n → a and b_n → b.

Since a_n → a, for a given ꜫ > 0, there is a positive integer m₁ such that | a_n – a | < ꜫ/2 for all n ≥ m₁

Since b_n → b, for a given ꜫ > 0, there is a positive integer m₂ such that | b_n – b | < ꜫ/2 for all n ≥ m₂

Put m= max { m₁, m₂ }

∴ | a_n – a | < ꜫ/2 and | b_n – b | < ꜫ/2 for all n ≥ m

Put z= a+ib.

Now |z_n – z | = | ( a_n +ib_n ) – (a+ib) |

= | (a_n -a) + i ( b_n -b) |

≤ | a_n – a | + |b_n – b |

< ꜫ/2 + ꜫ/2 = ꜫ.

∴ | z_n – z | < ꜫ for all n ≥ M

Thus { z_n } converges to z in the complex plane C.

Hence the complex plane is a complete metric space.