Notation for a subsequence of a sequence If we have a sequence (an ordered list)
$$
S=(s_0,s_1,...,s_n).
$$
What is the notation for expressing that $S'$ is a (ordered) subsequence of $S$?

Here is a handout on sequences that manages to avoid the problem by stating the definition of a subsequence and then simply stating that one is a subsequence of another each time. This, however, does not seem to be what one would like to do.
Here instead is a is another handout by Dr. Carryn Bellomo Warren
 at University of Nevada, Las Vegas.
This uses and explains the notation that David Mitra.
$$S'=(x_{n_{k}})_{n \in N} \ \ \text{for} \ n_1< n_2<n_3\cdots$$
And if you want to denote this just with the $S'$, from Structural Additive Theory, by David J. Grynkiewicz, we find the notation of defining a subsequence $S' | S$, where $S$ is a sequence. He also notes in his book that "All notation and conventions for [unordered] sequences extend naturally to ordered sequences."
Hopefully you are now convinced that this notation is relatively well used, and you should be able to find plenty more examples of this by searching online for "sequences notation". One thing to note is that oftentimes these curly brackets {} will be substituted for parentheses (), such as here, or in Mitra's comment itself.
