Second Subsequence Rule

Theorem
Let $M = \struct {A, d}$ be a metric space.

Let $\sequence {x_n}$ be a sequence in $M$.

Suppose $\sequence {x_n}$ has a subsequence which is unbounded.

Then $\sequence {x_n}$ is divergent.

Proof
Follows by the Rule of Transposition from Convergent Sequence is Bounded.

Also see

 * First Subsequence Rule