Subset of Naturals is Finite iff Bounded

Theorem
Let $X$ be a subset of the naturals.

Then $X$ is finite if and only if it is bounded.

Proof
Since the set of natural numbers is bounded below, $X$ is limited if and only if there exists $p\in\N$ such that $x\leq p$ for all $x\in X$.

Well, if $X=\left\{x_1, x_2, \ldots, x_n\right\}$ is finite, let $p=x_1+x_2+\ldots+x_n$. Then we see that $x\in X\implies x\leq p$, hence $X$ is limited.

On the other hand, if $X\subset\N$ is limited, then it is contained in $\N_p$ for some $p\in\N$. It follows from Subset of Finite Set is Finite that $X$ is finite.