TITLE: Supremum of a sequence of functions
QUESTION [5 upvotes]: I have a problem understanding the function $\sup f_n(x)$ on a domain of convergence. Given a sequence of functions $f_n(x)$ I guess that $\sup f_n(x)$ is the function such that for each x we find the lub of the values of all $f_n$ on $x$. 
To make thinks more simple, say that the functions $f_n$ are all bounded,then $\sup f_n$ means that for each $x$ of the domain of definition and convergence one finds the max of the values of $f_n$ on the particular $x$. Am I right ?
Now I present an example:
$$\sup\{|x^k|: -1 < x < 1, k \in \mathbb{N} \} = 1$$ which I don't understand. If I follow the thinking scheme of above, I should find $|x|$ and not $1$.
So where is the difficulty? Is my understanding of the definition of supremum of a sequence of functions wrong?
Thanks for any comments.

REPLY [6 votes]: In general, we take suprema of sets of real numbers, which is written $\sup S$ for a subset $S\subseteq \mathbb R$. Notations $\sup_{n\in\mathbb N}a_n$ can be rewritten as $\sup\{\,a_n: n\in\mathbb N\,\}$. Likewise $\sup_{n\in\mathbb N}f_n(x)=\sup\{\,f_n(x): n\in\mathbb N\,\}$ where the set on the right depends on the parameter $x$, so that the whole supremum-expression becomes a function of $x$. Your final example
$$\sup\{\,|x|^k: -1\le x\le 1, k\in\mathbb N\,\} $$
is already given with a subset of $\mathbb R$; the $x$ occuring in it does not make us consider functions for $x$ (like $k$) is not a parameter (or "free varaiable") here. As cleraly $|x|^k\le 1$ for all $x,k$ with $-1\le x\le 1$ and $k\in\mathbb N$, we see that $1$ is an upper bound for the set; and as $1$ actually is in the set, we have thus found th eleast upper bound ...