Convergent Sequence with Finite Number of Terms Deleted is Convergent

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

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

If $\left \langle {x_k} \right \rangle$ converges, the deletion of a finite number of terms will still result in a convergent sequence.

Similarly, if $\left \langle {x_k} \right \rangle$ diverges, the deletion of a finite number of terms will still result in a divergent sequence.