First: the "least upper bound" and "greatest lower bound" principles are not at all about ordered fields, they are about partially ordered sets; an ordered field is necessarily a partially ordered set, but there are plenty of partially ordered sets that satisfy the least upper bound and greatest lower bound properties without being fields.
Second: You are trying to prove the least upper bound principle, assuming the greatest lower bound principle holds. Let's be clear about what these principles are:
We say a partially ordered set $(X,\leq)$ satisfies the Greatest Lower Bound Principle if and only if every nonempty subset $P$ of $X$ that has a lower bound has a greatest lower bound; that is, if there exists $x\in X$ such that $x\leq p$ for all $p\in P$, then there exists $g \in X$ such that:
- $g\leq p$ for all $p\in P$; and
- if $x\in X$ is such that $x\leq p$ for all $p\in P$, then $x\leq g$.
As for the Least Upper Bound Principle, we have the dual definition:
We say a partially ordered set $(X,\leq)$ satisfies the Least Upper Bound Principle if and only if every nonempty subset $P$ of $X$ that has an upper bound has a least upper bound; that is, if there exists $x\in X$ such that $p\leq x$ for all $p\in P$, then there exists $\ell\in X$ such that:
- $p\leq \ell$ for all $p\in P$; and
- if $x\in X$ is such that $p\leq x$ for all $p\in P$, then $\ell\leq x$.
Now, let's look at your argument. You are assuming that your field $F$ satisfies the Greatest Lower Bound Principle, and you take a subset $P$ of $F$. You assume it is nonempty, and that there is an element $k$ that is an upper bound. You incorrectly assert that this element $k$ will be in $P$: you don't know that, and you don't need that.
Then you consider the set $-P=\{-p \mid p\in P\}$, and note that $-k$ is a lower bound for $-P$ (you incorrectly assert it is a greatest lower bound; you don't know that). You then deduce that $-P$ has a greatest lower bound (correct). And you stop there. But you need to show that $P$ has a least upper bound, and you have not done so. (Of course, you probably meant to take the greatest lower bound $\ell$ of $-P$, and then show that $-\ell$ is a least upper bound for $P$; but you have not done so). So your argument is incomplete.
Now, there is nothing wrong with the idea behind your argument (provided you fix the errors and fill in the gaps). The reason you may not want to do this is that the fact that the Least Upper Bound Principle and the Greatest Lower Bound Principles are equivalent is true for any partially ordered set. In a partially ordered set, the argument that relies on "multiplying by $-1$" will not in general work, because that operation will not make sense. So rather than use a proof that only works for a certain kind of partially ordered set, we can use a proof which is not any more complicated, but which works in general. And moreover, a proof that highlights that the least upper bound of a set is in fact a greatest lower bound of a different set (and likewise for greatest lower bounds).
Namely:
Suppose $X$ satisfies the greatest lower bound principle, and we want to prove it satisfies the least upper bound principle. Let $P$ be a nonempty set that has at least one upper bound. We need to show it has a least upper bound. Let $B=\{b\in X\mid b\text{ is an upper bound for }P\}$. By assumption, $B$ is not empty. Moreover, since $P$ is not empty, there is a $p\in P$, and so $p\leq b$ for all $b\in B$. That means that $B$ is a nonempty set that is bounded below. By the Greatest Lower Bound Principle, $B$ has a Greatest Lower Bound; call it $g$. We claim that $g$ is also a least upper bound for $P$.
First, note that if $p\in P$ we have $p\leq b$ for all $b\in B$. By the second property of the greatest lower bound, we conclude that $p\leq g$. Therefore, for every $p\in P$ we have $p\leq g$, so $g$ satisfies the first property necessary to be the least upper bound of $P$. To verify it satisfies the second property, if $x\in X$ is such that $p\leq x$ for all $p\in P$, then $x\in B$ by definition. And since $g$ is the greatest lower bound of $B$, then the first property of the greatest lower bound tells us that $g\leq b$ for all $b\in B$, and in particular that $g\leq x$. Thus, if $x$ is an upper bound for $P$, then $g\leq x$. This verifies that $g$ satisfies the second property of the least upper bound for $P$. We conclude that $g$ is indeed the least upper bound for $P$.
So we have shown that if $X$ satisfies the greatest lower bound principle, then any nonempty subset that is bounded above has a least upper bound; that is, $X$ also satisfies the least upper bound principle.
A symmetric argument (or applying the above argument to the partially ordered set $(X,\leq^{\rm op})$) shows that if $X$ satisfies the least upper bound principle, then it satisfies the greatest lower bound principle. $\Box$
Once you fill in all the details into your argument, you'll see that it is no shorter and no less difficult than the above one. But the above one has the virtue of not needing any of the field properties that you use (that $a\leq b$ if and only if $-b\leq -a$, for example), only the properties of order, of upper and lower bounds, and the corresponding principle. So it is a nicer proofs, because it assumes less, concludes the same, and is not any harder or longer than the proof you suggest.
Best Answer
The first part of your proof looks good! Nice choice of $\varepsilon.$ That establishes that $S$ has a least upper bound $b$ and $(1)$ shows that $b \leq 5$.
There are some bumps on the second half but you are definitely on the right track. First, to be pedantic, looks like you used $r$ on two occasions instead of $\varepsilon$. No biggie. The real problem is with your reasoning about $b$. Note that $b$ is still acting as the least upper bound for $S$, and it is perfectly fine for a least upper bound of a set to be an element of the set itself. For example, consider the closed unit interval $[0,1]$. It's least upper bound is $1$, a member of $[0,1]$.
That being said, $(2)$ was executed excellently and still completes the proof. The reasoning is that $(2)$ reveals a number $b+\varepsilon \in \Bbb{R}$ such that $b^3<(b+\varepsilon)^3<5$. Hence $b+\varepsilon \in S$, so the contradiction is that you've found an element in $S$ larger than the least upper bound of $S$.
P.S. You ask great questions. Complete with $\LaTeX$, a clear summary of what you've tried so far and you respond to input from other MSE users. Keep it up!