Q:

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

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.  

A:

$I$ 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 $I$.

