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 limits.

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 limits, it must in fact be divergent.