Is product of two decreasing functions (and positive) decreasing as well?

Question

I have two functions $f(x)>0$ and $g(x)>0$, both decreasing. Then, can I claim that the product $f(x)g(x)$ is decreasing as well? ($x>0$)

Answer 1

Yes, $0<a<b$ and $0<c<d$ implies $ac <bc <bd$ so $ac <bd$.

Let $x<y$ and take $a=f(x), b=f(y), c=g(x), d=f(y)$.

Answer 2

Assume $y>x$. We have $f(x)\geq f(y)$, and since $g(x)>0$ it follows that $f(x)g(x)\geq f(y)g(x)$. Also, we have $g(x)\geq g(y)$ and since $f(y)>0$ we get $g(x)f(y)\geq g(y)f(y)$. Combining this together we indeed get $f(x)g(x)\geq f(y)g(y)$.