We prove that every convergent sequence is a Cauchy sequence. Convergent sequences are Cauchy, isn't that neat? This is the first half of our effort to prove that a sequence converges if and only if it is Cauchy. Next we will have to prove that Cauchy sequences are convergent! Subscribe for more real analysis lessons!

Proof of Triangle Inequality Theorem: [ Ссылка ]
Intro to Cauchy Sequences: [ Ссылка ]
Cauchy Sequences are Bounded: [ Ссылка ]
Proof that Cauchy Sequences Converge: [ Ссылка ]

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