Putnam 2012, Problem B-1

Let $S$ be a class of functions from $[0, \infty)$ to $[0, \infty)$ that satisfies:  
(i) The functions $f_1(x) = e^x - 1$ and $f_2(x) = \ln(x+1)$ are in $S$;  
(ii) If $f(x)$ and $g(x)$ are in $S$, the functions $f(x) + g(x)$ and $f(g(x))$ are in $S$;  
(iii) If $f(x)$ and $g(x)$ are in $S$ and $f(x) \geq g(x)$ for all $x \geq 0$, then the function $f(x) - g(x)$ is in $S$.  
Prove that if $f(x)$ and $g(x)$ are in $S$, then the function $f(x)g(x)$ is also in $S$.

Solution:
Each of the following functions belongs to $S$ for the reasons indicated.

$f(x), g(x)$ given

$\ln(x+1)$ (i)

$\ln(f(x)+1), \ln(g(x)+1)$ (ii) plus two previous lines

$\ln(f(x)+1) + \ln(g(x)+1)$ (ii)

$e^x - 1$ (i)

$(f(x)+1)(g(x)+1) - 1$ (ii) plus two previous lines

$f(x)g(x) + f(x) + g(x)$ previous line

$f(x) + g(x)$ (ii) plus first line

$f(x)g(x)$ (iii) plus two previous lines