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 bounded iff $X$ is bounded above.

That is, there exists $p \in \N$ such that $x \leq p$ for all $x \in X$.

Let $X$ be finite.

Then $X = \left\{{x_1, x_2, \ldots, x_n}\right\}$ for some $n \in \N$.

Let $p = x_1 + x_2 + \ldots + x_n$.

Then we see that $x \in X \implies x \leq p$, hence $X$ is bounded.

Now let $X \subset \N$ be bounded.

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.

$\blacksquare$

Sources

• Elon Lages Lima: Análise Real 1 (1989): $\text{Chapter 1}, \S 2$