# Concept derivation guides

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Concept guide



## [Cauchy sequence: a complete certificate for the reciprocal sequence, a space in which it has no limit, an alternating counterexample and an ultimately constant model](/en/derivations/derivation-mko-cauchy-sequence-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-cauchy-sequence)

(1) Prove that x_n=1/n (n≥1) is a Cauchy sequence in (ℝ, |s−t|): for every ε>0 give N such that |1/m−1/n|<ε for all m,n≥N. (2) Prove that the same sequence is still Cauchy in X=(0,1] with the restricted metric, but has no limit in X. (3) Prove that y_n=(−1)^n is not Cauchy. (4) Prove that the ultimately constant sequence z (z₁=0, z₂=7, z_n=5 for n≥3) is Cauchy. Also explain why a finite tail cannot prove the Cauchy property.

Cauchy sequence

derivation-mko-cauchy-sequence-01





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