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

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

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

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

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

**Kane, Jonathan**: "Writing Proofs in Analysis", Springer, 2016