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Proof

(related to Corollary: Non-Cauchy Sequences are Not Convergent)

Context

• Let $(a_n)_{n\in\mathbb N}$ be a real sequence.

Hypothesis

• Let $(a_n)_{n\in\mathbb N}$ be not a Cauchy sequence.

Implications

• Since all Cauchy sequences converge, it follows that $(a_n)_{n\in\mathbb N}$ cannot converge, by contraposition.

Conclusion

• Thus, $(a_n)_{n\in\mathbb N}$ is not convergent.
  ∎

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References

Bibliography

1. Kane, Jonathan: "Writing Proofs in Analysis", Springer, 2016