Let
x, y be elements ofR such that x not= y. Prove that there exists epsom > 0 such that y not element of[x − epsom, x + epsom]
Assume $\displaystyle y\in[x-\frac{d(x,y)}{2}, x+\frac{d(x,y)}{2}]$
Then $\displaystyle x-\frac{d(x,y)}{2}\leq y\leq x+\frac{d(x,y)}{2}$
So $\displaystyle -\frac{d(x,y)}{2}\leq y-x\leq\frac{d(x,y)}{2}$
On the real line, $\displaystyle d(x,y)=|x-y|$ so
$\displaystyle -\frac{1}{2}|x-y|\leq y-x\leq \frac{1}{2}|x-y|$
So $\displaystyle -\frac{1}{2}|x-y|\leq y-x$ and
$\displaystyle y-x\leq\frac{1}{2}|x-y|$
Multiplying by -2 and 2 respectively yields
$\displaystyle -2(y-x)\geq |x-y|$ and
$\displaystyle 2(y-x)\geq |x-y|$
These two facts imply x=y, but that is a contradiction with the given information