(related to Theorem: Completeness Principle for Real Numbers)
If a real sequence $(a_n)_{n\in\mathbb N}$ is not a Cauchy sequence, then it does not converge.
If $(a_n)_{n\in\mathbb N}$ is not Cauchy, there is some $\epsilon > 0$ and a natural number $N(\epsilon)$ such that $|a_{m}-a_n|\ge \epsilon$ for all $m,n > N(\epsilon).$
Thus, to prove that a sequence is not convergent, finding corresponding numbers $\epsilon > 0$ and $N(\epsilon)$ such that $|a_{m}-a_n|\ge \epsilon$ for all $m,n > N(\epsilon)$ will do.
Proofs: 1