Strictly Increasing Infinite Sequence of Positive Integers is Cofinal in Natural Numbers

Definition
Let $S = \left\langle{x_n}\right\rangle$ be an infinite sequence of positive integers which is strictly increasing.

Then $S$ is a cofinal subset of $\left({\N, \le}\right)$ where $\le$ is the usual ordering on the natural numbers.

Proof
By definition, $S = \left\langle{x_n}\right\rangle$ is a subset of $\N$.

The result follows from Subset of Natural Numbers is Cofinal iff Infinite.