Second Subsequence Rule

Theorem

Let $M = \left({A, d}\right)$ be a metric space.

Let $\left \langle {x_n} \right \rangle$ be a sequence in $M$.

Suppose $\left \langle {x_n} \right \rangle$ has a subsequence which is unbounded.

Then $\left \langle {x_n} \right \rangle$ is divergent.

Proof

Follows directly from the result that a Convergent Sequence is Bounded.

$\blacksquare$