Is the following statement true? And if so where can I find a proof

Question

Is the following true for real numbers?

If $x < a*b$ then there exists $c$ and $d$ such that $x=c*d$ and $a>c$ and $b>d$.

Thanks...

Answer 1

Suppose $x=a\times b - \epsilon$, with $\epsilon>0$. We now want to find positive $\epsilon', \epsilon''$ such that $$a\times b-\epsilon=(a-\epsilon')\times (b-\epsilon'')=a\times b - b\epsilon' -a\epsilon''+\epsilon'\epsilon''$$

We can do this; in fact, we can even insist that $\epsilon'=\epsilon''=\delta$. Then the equation we need to solve is $$\delta^2-(a+b)\delta+\epsilon=0$$

This has solutions, using the quadratic formula, as $$\delta =\frac{a+b\pm\sqrt{(a+b)^2-4\epsilon}}{2}$$ Taking the + version we will be assured that $\delta>0$ (assuming $a,b>0$).

Answer 2

$5<2*3$

$x=-1*(-5)$

now clearly $2>(-1)$ and $3>(-5)$

clearly it exist,unless we say that $c$ and $d$ must be positive numbers or we make some general constraint on numbers