Let $A$ be a nonempty, bounded (from above) set consisting of non-negative real numbers only, let $n$ be a given positive integer, and let sets $B$ and $C$ be defined as follows:$$B := \left\{ a^n \colon a \in A \right\} \qquad \mbox{ and } \qquad C := \left\{ \sqrt[n]{a} \colon a \in A \right\}. $$Then obviously $B$ and $C$ are also nonempty and bounded. How to rigorously show that$$\sup B = \big( \sup A \big)^n \qquad \mbox{ and } \qquad \inf B = \big( \inf A \big)^n?$$And, how to show rigorously that$$\sup C = \sqrt[n]{ \sup A } \qquad \mbox{ and } \qquad \inf C = \sqrt[n]{ \inf A }?$$
My Attempt:
We show the result about the supremum first. Obviously, $\big( \sup A \big)^n$ is an upper bound for our set $B$ so that $B$ is bounded from above.
How to proceed from here?
Can we prove the required results using only the material in Chapter 1 of Baby Rudin, for example?