Intuition for sequences of functions? A sequence $(a_n)$ of real numbers can be thought of as a function that maps $\mathbb{N}$ to $\mathbb{R}$. The supremum of this sequence, if it exists, will be some $k \in \mathbb{R}$.
A regular function $f$ from $\mathbb{R} \to \mathbb{R}$ will also have supremum of some $k \in \mathbb{R}$, if this supremum exists.
Now, what about a sequence of functions $(f_n)$. It seems that here we have a mapping from $\mathbb{N}$ into a range that consists of functions. Which, will in turn map values from $\mathbb{R}$ into $\mathbb{R}$.
So what is the supremum of a sequence of functions? Is it a function itself, what does it 'look like' intuitively in relation to the other functions in $(f_n)$?
Or is the supremum some $k \in \mathbb{R}$? As in the smallest upper bound of all the functions in $(f_n(x))$ over all $x \in \mathbb{R}$?
 A: A sequence of functions can be thought of as a mapping $g:\mathbb{N}\to\mathbb{R}^{\mathbb{R}}$ defined by $g(n)=f_{n}$ where $\mathbb{R}^{\mathbb{R}}$ is the space of functions that map $\mathbb{R}$ into $\mathbb{R}$. The supremum will be the map defined by $f(x)=\sup_{n\in\mathbb{N}}f_{n}(x)$. This may not be a function in $\mathbb{R}^{\mathbb{R}}$ since $\sup_{n\in\mathbb{N}}f_{n}(x)$ may not be in $\mathbb{R}$ for some $x\in\mathbb{R}$ (or any $x$ in fact). This function will be the smallest upper bound on the sequence of functions when it exists. That is if $f_{n}\le g$ for all $n\in\mathbb{N}$ then $f\le g$. This follows since $g$ being an upper bound on all $f_{n}$ means the real number $g(x)$ is an upper bound on the sequence $f_{n}(x)$. Hence, $f(x)=\sup_{n\in\mathbb{N}}f_{n}(x)\le g(x)$.
A: Note that the supremum is an operator on a set, in other words when we consider sequences we actually look at
$$
\sup \{ a_n \mid n\in \mathbb{N} \}
$$
And similarly for functions. The usual way I have seen for defining $\sup_{n \in \mathbb{N} } f_n$ where each $f_n$ is a function is to define a new function $f$ Point wise like so
$$
f(x) = \sup_n f_n(x)
$$
However note that we can (although I have never seen this used other than for my own curiosity) define $\tilde{f} \in \overline{\mathbb{R}}$ like so
$$
\tilde{f} = \sup_{ n } \sup_{x \in \mathbb{R}} f_n(x)
$$
That is, for each function in the sequence look at the supremum of it, then make a sequence of these extended real numbers and take a look at the sup of those. This, in some sense will give the largest possible value of all the functions combined.
EDIT: I wanted to verify this fact before I posted it, but in fact we have that $\tilde{f} = \sup_{x \in \mathbb{R}} f(x)$, the proof isn't too difficult and I could post it here if wanted. Also note something that I have used before is
$$
\limsup_{n \to \infty} \sup_{x \in \mathbb{R}} f_n(x)
$$
To see the limiting largest value of these functions (in the sense that it approaches these values of course).
