0
$\begingroup$

How to prove $|[a, b]| = |\mathbb R|$, i.e. there are equally many points on a finite line segment than on the whole number line? Could one use the Cantor-Schröder-Dedekind theorem, which tells as that if we have injections $A\to B\to A$, then $|A| = |B|$? Then it only remains to prove that there is an injection $\mathbb R\to [a, b]$.

$\endgroup$
2
1
$\begingroup$

It’s easy to get a one-to-one onto map between $\Bbb R$ and an open interval $\langle a,b\rangle$, you can use $\arctan$ for instance, suitably scaled, to do it. Then there are standard tricks to get the endpoints into the act. Nothing as deep as Schröder-Bernstein should be necessary.

$\endgroup$
2
  • 1
    $\begingroup$ I'm not sure I would call Schröder-Bernstein particularly deep, though. $\endgroup$ – hmakholm left over Monica Feb 22 '17 at 23:29
  • $\begingroup$ For my level of smarts, @HenningMakholm, it’s deep enough. $\endgroup$ – Lubin Feb 23 '17 at 14:07
4
$\begingroup$

There is an injection (bijection, in fact) from $\mathbb R$ to $(-1,1)$: $$f(x)=\frac{x}{1+|x|}$$

Now, transform the interval $(-1,1)$ into $(a,b)$.

$\endgroup$

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