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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.

Best regards,

lorenzo.

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Drawing a picture helps with this one. For one case, let's assume $y \leq w \leq z \leq x$.
$$x - w \leq x - y$$ So then $|x - w| \leq |x - y| \leq \epsilon$. You can then run through the other possibilities in a similar way.

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