Power of Real Number greater than One is Unbounded Above

Theorem
Let $x \in \R$ be a real number such that $x > 1$.

Let set $S = \left\{{x^n: n \in \N}\right\}$.

Then $S$ is unbounded above.

Proof
Suppose $S$ were bounded above.

Then $S$ has a supremum $B$.

As $x > 1$, it follows that $\dfrac B x < B$ and so therefore $\dfrac B x$ can not be an upper bound.

Therefore:
 * $\exists n \in \N: x^n > \dfrac B x \implies x^{n+1} > B$

So $B$ can not be an upper bound.

From that contradiction it can be concluded that $S$ can not have an upper bound and therefore $S$ is unbounded above.