Boundedness of Nth Powers

Theorem

Power of Real Number greater than One is Unbounded Above

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

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

Then $S$ is unbounded above.

Power of Real Number between Zero and One is Bounded

Let $x \in \R$ be a real number.

Let $0 < x < 1$.

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

Then:

$\inf S = 0$

and:

$\sup S = 1$

where $\inf S$ and $\sup S$ are the infimum and supremum of $S$ respectively.