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 $\mathbb R\to [a, b]$.
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.