Here is what I've done so far:
[First, want to show $b = 5$ is an upper-bound of $S$.]
So, let: $$S = \{x \in \Bbb R : x \gt 0, x^3 \le 5\}, S \neq \emptyset$$
Assume that $b = 5$ is not an upper-bound of $S$. Then, $\exists x \in S$ s.t. $x^3 \gt 5$. But this contradicts the definition of $S$. Therefore, $b = 5$ is an upper-bound of $S$.
[Next, want to show if $c \lt b$, then $c$ is not an upper-bound.]
This is where I'm stuck… Are you supposed to pick an epsilon here? I tried this also:
Let $\epsilon \gt 0$, $\epsilon = {b – c\over 2}$. I'm not sure how to proceed from here on.
Best Answer
The set is non-empty because $1\in S$, so we have a non-empty subset of $\mathbb R$ that has an upper bound then $S$ has a supremum, say $\alpha$, Claim: $\alpha^{3}=5$, try to prove the claim.