Divergent Sequence with Finite Number of Terms Deleted is Divergent

Theorem
Let $\left({X, d}\right)$ be a metric space.

Let $\left \langle {x_k} \right \rangle$ be a sequence in $X$.

Let $\left \langle {x_k} \right \rangle$ be divergent.

Let a finite number of terms be deleted from $\left \langle {x_k} \right \rangle$.

Then the resulting subsequence is divergent.

Also see

 * Tail of Convergent Series tends to Zero