I know this is kind of minor. But I want to know if I understand this correctly. In "baby rudin", on page 42, is the condition
$$3^{-m}\lt \frac{\beta-\alpha}{6}$$
tight? I thought $$3^{-m}\lt \frac{\beta-\alpha}{4}$$ is sufficient, am I correct?
Thanks.
Edit: Okay, the background is: the Cantor set (i.e. the union of all intervals after repeatedly removing all the middle third interval) does not contain any point lying in the segment
$(\frac{3k+1}{3^m},\frac{3k+2}{3^m})$
And he was saying every segment $(\alpha, \beta)$ contains a segment of the form above, if
$$3^{-m}\lt \frac{\beta-\alpha}{6}$$
But I thought this condition is too loose.