Prove that there is a real number $r>0$ such that:
There is no point in $\mathbb{R}^3$ with 3 rational coordinates, whose distance from $(0,0,0)$ equals $r$. In other words, if we build a sphere with radius $r$ and center in $(0,0,0)$, it will contain no points with 3 rational coordinates.
It sounds logical, and I think there are a couple of such numbers, but how can I write a fully formal proof of this? I guess it'll be sufficient to prove just one example, and that's what I tried:
for $r=\sqrt[4]{2}$
$x^2+y^2+z^2=\sqrt{2}$
A sum of three rational numbers can't be irrational. But is this enough of a proof? If not, can someone point me in the right direction? Also, this is a homework for elementary math (elementary set theory, relations, orders etc), so I fear it can't be that simple...
Thanks!
