Second Subsequence Rule
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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.
$\blacksquare$