What is the correct way to express the idea that a set is "entirely contained" in a given interval?

For example:

The set $ \ $ $S= \{1, 2, 3 \}$ $ $ is in some sense "contained" in the interval $ \ $ $I=[0, 4]$ $ $ since every element of $ $ $S$ $ $ belongs to $I$. $ \ $ Still, this wording sounds clumsy and may even be nonsense.

How should I express this idea properly and concisely?

I'm thinking of expressing $I$ as some set $T$, then I could write $S \subset T$. $ \ \ $ Still, this is not exactly what I want to claim, not to mention that some additional work might be needed to express $I$ as a set.

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    $\begingroup$ Intervals are sets. You don't need to rename it to be able to write $S\subset I$ $\endgroup$ – user545497 Mar 27 '18 at 12:12
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    $\begingroup$ But $I$ is already a set. By definition, $[a,b]=\{x\in\mathbb R| a\leq x\land x\leq b\}$. $\endgroup$ – 5xum Mar 27 '18 at 12:12
  • $\begingroup$ @5xum I see, so the way we write intervals is just a particular way of writing a set when it's a subset of $\mathbb{R}$? $\endgroup$ – Stephen Mar 27 '18 at 12:19
  • $\begingroup$ @Stephen Not only a subset of $\mathbb R$, but a very specific subset of $\mathbb R$, one defined as "all numbers between two numbers". $\endgroup$ – 5xum Mar 27 '18 at 12:19
  • $\begingroup$ Great, makes sense. $\endgroup$ – Stephen Mar 27 '18 at 12:21

$I$ is already a set. By definition, $[a,b]=\{x\in\mathbb R| a\leq x\land x\leq b\}$.

So what you want to express is written simply as $S\subseteq I$, since that means that every element of $S$ is also an element of $I$.

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