I am fairly sure this question will look quite trivial, but I have some trouble getting the reason behind the use of the symbole $\subset$ when we deal with sequences and initial segments of sequences. Indeed, to me this looks kind of – if not improper – at least potentially misleading.
Just few preliminaries to agree on the topic and notation.
W.l.o.g. let $A$ be a countable set. then $A^\mathbb{N}$ be the set of sequences on $A$, i.e. the set of functions that map $\mathbb{N}$ to $A$. Thus, given $\sigma, \tau \in A^\mathbb{N}$, we write $\sigma \subset \tau$ to indicate that $\sigma$ is an initial segment of $\tau$.
As it stands now, this looks to me not completely precise. Indeed, if we distinguish (and we do!) between $\subset$ and $\subseteq$, by writing $\sigma \subset \tau$, then $\sigma \in A^{<\mathbb{N}}$ and $\tau \in A^\mathbb{N}$. Thus, $\sigma$ looks like $(a_1, a_2, \dots, a_n)$, where $\tau$ looks like $(a_1, a_2, \dots)$. But then $\sigma$ is not really a subset of $\tau$.
Anyway, beyond this fact, just to get a confirmation, Is actually the case that in general $\sigma$ is a subset of $\tau$ and I don't see it, or it is simply what I wrote before, i.e. $\sigma$ is an initial segment of $\tau$, and by chance we use to express that notion the same symbol that we use for the set inclusion?
Thank you for your feedback and for your time.
PS: As I wrote, this question can look trivial, but when somebody is self-taught as I am, those little things that are written nowhere are exactly the one that are problematic (another typical example is the fact that in math, definitions are written only by using if, and not the proper if and only if. It took me a while and some lecture notes found somewhere to get this convention)
If you take the set-theoretic view that a sequence $(a_1,\ldots,a_n)$ in $A$ is just a function $a : \{1,\ldots,n\}\to A$, and that a function is just a set of ordered pairs, then actually any initial segment of a sequence is a subset of that sequence, since the sequence $(a_1,\ldots,a_n)$ is equal to the set $$\{(1,a_1),\ldots,(n,a_n)\}$$whereas the infinite sequence $(a_1,\ldots,a_n,a_{n+1},\ldots)$ is equal to the set $$\{(1,a_1),\ldots,(n,a_n),(n+1,a_{n+1}),\ldots\}$$
Edit: I should point out that there are subsets which are not initial segments (e.g. the subset $\{(2n,a_{2n}) : n\in\mathbb{N}\}$), though every subset is a subsequence in a certain sense. I think some set-theorists use an alternate notation to $\subseteq$ to indicate that a sequence is an initial segment, and not just a subsequence.