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.