1. Cauchy Criterion

Definition (Convergence sequence).

A sequence of real numbers is said to be converges to a real number if for every , there exist such that

Definition (Cauchy Sequence ).

A sequence of real number is said to be Cauchy if for every , There exist such that

Theorem (Bolzano-Weierstrass).

Every bounded sequence of real numbers has a convergent sequence.

Theorem (Convergent Sequences are Cauchy).

Let be a convergent sequence of real number. Then is a Cauchy.

Proof.
suppose converges to
for every such that

Now for

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Lemma (Cauchy Sequences are Bounded).

let be a Cauchy sequence
Then it is bounded

Proof.
Take As is Cauchy There exist such that

take Then,

That is
so
then where
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Theorem (Every Cauchy Sequence is Convergent).

Let be a Cauchy sequence of real number. Then is convergent

Proof.

By the theorem Cauchy Sequences are Bounded is bounded. Then by Bolzano-Weierstrass Theorem has a convergent sequence.
Suppose is a Convergent sequence
Take
For such that

such that

Take

Then by(1.2)
now for

from(1.1)


Which implies Cauchy sequence is convergent
□

Theorem.

A sequence of real numbers id Cauchy iff it is convergent

Let be a non-empty set
Suppose is a function satisfying

  1. (positivity)
  3. (symmetry)
  4. (Triangle inequality)

Suppose is a metric space then with is said to be convergent to if converges to
A sequence is said to be Cauchy if for there exist such that

Definition (Complete Metric Space).

A metric space is said to be complete if every Cauchy sequence in converges

Examples

Let

  1. is complete.
  2. Let consider the sequence it is Cauchy sequence but not convergent in so, its not complete
  3. is not complete
  4. is complete
  5. is complete
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