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SEQUENCES
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THEOREM
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Cauchy and Convergence
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Introduction Sequences
Subsequences
Subsequences Convergents
Bounded Sequences
Monotonic Sequences
Examples Sequences 1
Converge of a Sequence
Examples Sequences 2
Limit Uniqueness
Convergence and Boundness
Bounded and not Converges
Permanence of the Sign
Cauchy Sequence
Cauchy Condition
Cauchy and bounded
Convergent Condition
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Cauchy and Convergence
Bounded and Convergence
If $\{s_n\}$ is a Cauchy sequence, then it is convergent.
PROOF
We have that it is bounded (by lemma
Cauchy and Bounded
), also it has a convergent subsequence because it is bounded and by lemma
Convergent condition
, it is convergent.