Subsequence Rules

Theorem

First Subsequence Rule

Let $T = \struct {A, \tau}$ be a Hausdorff space.

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

Suppose $\sequence {x_n}$ has two convergent subsequences with different limits.

Then $\sequence {x_n}$ is divergent.

Second Subsequence Rule

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.