I'm studying Analysis from Terence Tao's book 'Analysis 1' and in an exercise he asks to prove seven properties regarding the notion of '$\varepsilon$ - closeness', which is defined as follows:
Def: Let $\varepsilon>0$, $x$, $y$ be rational numbers. We say that $y$ is $\varepsilon$-close to $x$ iff we have $d(y,x)\leq \varepsilon$.
I've managed to prove six out of the seven properties; the only one which I haven't been able to prove is the following one:
"Let $\varepsilon>0$. If $y$ and $z$ are both $\varepsilon$-close to $x$, and $w$ is between $y$ and $z$ (i.e. $y\leq w\leq z$ or $z\leq w \leq y$), then $w$ is also $\varepsilon$-close to $x$."
So, I would appreciate any hints about proving this property.