First Subsequence Rule

Theorem
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 limit.

Then $\sequence {x_n}$ is divergent.

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

From Limit of Subsequence equals Limit of Sequence, any subsequence of a convergent sequence must have the same limit.

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