Can a sequence of a function with a single variable be thought about as a function with two variables? Long title, but first off is it logically ok to think of $\{f_n(x)\}$ as $f(n,x)$ where $n$ is restricted to a natural number?
Second, would this at all be useful? Thus far in my study of sequences of functions you deal strictly with convergence (similar to sequences in general) - either pointwise or universal convergence.
It seems interesting to me because then we can consider some function of two variables as converging to a function with one. It seems as though this would be useful...
So it's as if we could have the definition of convergence be:
$$
\forall \; \epsilon > 0 \\
\exists \; N \in \mathbb{N} : \lvert f(n,x) - f(x) \rvert < \epsilon
$$
and then could this possibly be extrapolated out to an $n$ with any domain one wishes?
Thank you in advanced, sorry if this is a silly question and I ask if you downvote, put in the comments why!
EDIT: as Andre reminded me, sequences can be thought of as functions, i.e.
$$
a : \mathbb{N} \to \mathbb{R}
$$
however it seems to be then that this becomes the concatenation of functions, i.e.
$$
g : \mathbb{R} \to \mathbb{R}
f : \{ \mathbb{R} , \mathbb{N} \} \to \mathbb{R}
$$
and $f_n (x) = f(g(x),n)$
again, would this be an ok way of thinking about sequences of functions, is it useful, and could this be extrapolated so that we can view any $n$ variable function as the concatenation of $n$ functions, and the variables dont need to have any specific domain?
 A: What you are describing can certainly be done and can be useful. It is akin to realizing that an ordinary sequence of real numbers is nothing but a function $\mathbb N \to \mathbb R$. So the study of sequences is subsumed by the theory of general functions. Moreover, given any space $X$ a sequence in $X$ is nothing but a function $\mathbb N \to X$, so now take $X$ to be the space of functions $\{f:\mathbb R \to \mathbb R\}$, then a sequence of elements in $X$ is nothing but a sequence $\{f_n(x)\}$ of single variable functions, and this amount to a function $F:\mathbb N \to X$, with $F(n)=f_n$. We may now define $G:\mathbb N \times \mathbb R \to \mathbb R$ by $G(n,x)=F(n)(x)=f_n(x)$, so yes a sequence of functions can be seen as a function to two variables. 
Now, to get a relation to limits you need to add a new points to $\mathbb N$ and topologize appropriately and, with some care, existence of the limits of a sequence becomes equivalent to the extendability of the corresponding function to this new point. This can be a useful way to look at things, though I don't think it is terribly useful. 
Lastly, instead of considering sequences you can consider wilder domains to replace $\mathbb N$ by. It is common for instance to consider nets, which is what happens if instead of the natural numbers with their natural ordering you allow any directed sets. In the theory of metric spaces, sequences suffice for all topological investigations of any metric space, but the same does not hold for arbitrary topological spaces, where nets become necessary.  
