I'm familiar with several proofs that the real numbers are uncountable (Cantor's initial proof, a proof by diagonalization, etc.). However, I've never seen a proof that the reals are uncountable that proceeds by showing that the set of Dedekind cuts of the rationals are uncountable. I'm aware that the set of all subsets of the rationals is uncountable, but not all of these sets are Dedekind cuts.
Is there a simple proof of the uncountability of $\mathbb{R}$ that works by showing the set of Dedekind cuts is uncountable?
Thanks!