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,



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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.