For $z =\frac{ xy}{x+y}$, why is $z < x$ and $z < y$ for all values of $x > 0$ and $y > 0$?

Question

This question has to do with the concept of resistors, given two resistors in parallel the equivalent resistance is always lower than the smallest individual resistance, I am trying to convince myself that this is true.

The equivalent resistance is given by $z$ and the individual resistances are $x$ and $y$.

Answer 1

Note that $$z={xy\over x+y} = x\cdot {y\over x+y}$$ and that ${y \over x+y}<1$ since the numerator is smaller than the denominator.

Answer 2

Because if $0<a<b$ then $\displaystyle\frac ca>\frac cb$. So, $$\frac{xy}{x+y}<\frac{xy}x$$

Answer 3

$x,y>0$ implies $$z=\frac{xy}{x+y}=\frac x{x+y}\cdot y<\frac{x+y}{x+y}\cdot y=y$$ and the same with $x$, $y$ interchanged.