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

Definition

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

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

Proof

By definition, $S = \sequence {x_n}$ is a subset of $\N$.

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

$\blacksquare$