First Subsequence Rule

Theorem

Let $T = \left({A, \vartheta}\right)$ be a Hausdorff space.

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

Suppose $\left \langle {x_n} \right \rangle$ has two convergent subsequences with different limit.

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

Proof

From Convergent Sequence in Hausdorff Space has Unique Limit, if $\left \langle {x_n} \right \rangle$ is convergent in a Hausdorff space it has exactly one limit.

From Limit of a Subsequence, any subsequence of such a sequence must have the same limit.

So, if a sequence has two convergent subsequences with different limit, it must in fact be divergent.

$\blacksquare$