How do I show there isn't an order isomorphism b/w the two sets $\{1, 2, 3,...\}$ and $\{1, 2, 3, ..., \omega \}$

Suppose that $f:\{1,2,3,\dots\}\to\{1,2,3,\dots,\omega\}$ is an order-preserving bijection. There must be some $n\in\{1,2,3,\dots,\}$ such that $f(n)=\omega$. What can $f(n+1)$ be?

Note that an order isomorphism preserves maximum properties, namely if $a$ is a maximum then $f(a)$ is a maximum of the image.

In particular a linearly ordered set with a maximum is never isomorphic to one without.