Definition of a bounded, nonempty subset of real numbers.
A bounded set is one which could be contained in an interval $[a,b]$
It could be finite or infinite, continuous or discrete.
For example the set $\{1,1/2,1/3,...\}$ is a nonempty bounded set because it is contained in $[0,1]$
The interval $(0,1)$ is bounded because it is contained in $[0,1]$
The set $\{1,2,3,4,5\}$ is bounded because it is contained in $[1,5]$
Recall that we can say $S \subseteq \Bbb R$ is "bounded" if there exists $m,M \in \Bbb R$ such that, for all $s \in S$, $m \le s \le M$. We also say $S$ is "nonempty" if ... well, it contains elements, that's simple enough.
Edit: a more intuitive idea of this principle is, as given by Mohammad Riazi-Kermani in their answer, is that $S$ is a nonempty subset of some finite interval. In my definition above, that interval is $[m,M]$.
Neither of these tenets require finiteness or connectedness. For example, $S = \{1\}$ is bounded, with $m=M=1$, and is obviously non-empty. The Cantor set is bounded and nonempty with $m=0,M=1$, but is totally disconnected. Your example set of $S=\{0,1,2,3\}$ is bounded and nonempty with $m=0,M=3$.
But we can also have continuous/connected examples as well. For instance, $S=(-2,2)$ is bounded and nonempty with $m=-2,M=2$. Or perhaps we take $S = (1,3) \cup (4,6)$, which is nonempty and bounded with $m=1,M=6$.
If I wanted a "continuous" subset, I would say "interval".
"$S = \{0, 1, 2, 3\}$"
Hmm...
- Bounded? Let's see if it's bounded by $100$. $|0| \leq 100$. $|1| \leq 100$. $|2| \leq 100$. $|3| \leq 100$. Yes, it's bounded.
- Nonempty? $1 \in S$, so $S$ is nonempty.
- Subset? $0 \in \Bbb{R}$, $1 \in \Bbb{R}$, $2 \in \Bbb{R}$, and $3 \in \Bbb{R}$, so $S \subset \Bbb{R}$.
Nonemptiness required a "choice", but that choice is trivial. The only one that involved a nontrivial "choice" was boundedness. Of course if you are picking a set, you probably know or can can probably find a (tight) bound. If the set is defined in a sufficiently complicated manner, finding a bound may require some work/creativity and such a set may be unbounded. (Consider the set of harmonic numbers. Looking at the small members, they might seem to be bounded. They aren't.)