Boundedness of Nth Powers

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Power of Real Number greater than One is Unbounded Above

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

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

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


$\inf S = 0$


$\sup S = 1$

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