We prove that a sequence converges if and only if it is Cauchy! This means that if a sequence converges then it is Cauchy, and if a sequence is Cauchy then it converges. Note that the definition of a Cauchy sequence has nothing to do with the particular limit of the sequence, so this gives us a way to prove a sequence converges without knowing its limit - we simply prove it is Cauchy! Before we prove either direction of this result we will prove that Cauchy sequences are bounded, since we use this result in the proof. With this proof done, we have got an awful lot of value from the Cauchy criterion for sequences!

Table of Contents
0:00 Intro
2:14 Every Cauchy Sequence is Bounded
7:17 Every Convergent Sequence is Cauchy
12:31 Every Cauchy Sequence is Convergent

#RealAnalysis #Math

Intro to Cauchy Sequences: [ Ссылка ]
Proof Cauchy Sequences are Bounded: [ Ссылка ]
Proof Every Convergent Sequence is Cauchy: [ Ссылка ]
Proof Sequence Converges if and Only if all of its Subsequences Do: [ Ссылка ]
Proof of Bolzano-Weierstrass Theorem (coming soon):
Proof Cauchy Sequences Converge: [ Ссылка ]

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