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

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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.

$\blacksquare$