set theory (concept of infinity) Let $S$ be a set. $f$ be a function on $S$ into real line $\mathbb{R}$. 
Let us define $Af$ as a function from $\mathbb{R}$ into power set of $S$, $PS$
$Af(x) = \{s\mid f(s) \leq x\}$ 
Question: Is $S$ in the range of $Af$? 
(as an example we can chose $S=\mathbb{R}$ and $f(x) = x$) 
Thanks
PS: I'm having some technical difficulties in proving few results. It boils down to above. (and related in infinity) 

$S$ is in the range of $Af$ iff $f$ is bounded from above.
If $f$ isn't bounded from above, there is for every $x\in \mathbb{R}$ a $s_x\in S$ with $f(s_x)>x$ and so $s_x\notin Af(x)$ and so $S\neq Af(x)$ which means $S$ isn't in the range.
If $f$ is bounded from above, there is a $x\in \mathbb{R}$ with $f(s)\leq x$ for all $s\in S$ and so $S=Af(x)$  which means $S$ is in the range.
