w > x and y > z then wy > xz
w,x, y and z are natural numbers
use only addition and multiplication for proof
I need help with this proof ASAP....please
The trouble here is that this is almost too obvious. The conclusion seems to follow immediately for natural numbers. So to make our proof seem more rigorous, let's take the long way around.
Proof
Let $\displaystyle w,x,y,z \in \mathbb{N}$.
Suppose $\displaystyle w \ge x$ and $\displaystyle y \ge z$. Then we have that $\displaystyle w - x \ge 0$ and $\displaystyle y - z \ge 0$. We now have two numbers that are non-negative, therefore, their product must be non-negative. Thus we have:
$\displaystyle 0 \le (w - x)(y - z) = wy - xz -xy - wz \Longleftrightarrow xz + (xy + wz) \le wy$.
Since $\displaystyle w,x,y,z \in \mathbb{N}$, $\displaystyle xy + wz \ge 0$. Thus we make the left side of the last inequality even smaller if we subtract it. Hence,
$\displaystyle wy \ge xz + (xy + wz) \ge xz \implies wy \ge xz$ as desired.
QED