So I guess we're looking for proof of $|\mathbb R|=2^{\aleph_0}$ or, in other words, $\mathfrak c=2^{\aleph_0}$.

The title of this looks like something close, but it explicitly asks for a proof using only axioms of ordered fields: Showing ${\frak c}:=|\Bbb{R}|=2^{\aleph_0}$ using only the axioms of the reals

Perhaps this: Easiest way to prove that $2^{\aleph_0} = c$?

This question seems to be about the same topic: Proof for the relationship between Cardinality of Natural numbers and Cardinality of Real Numbers

@Martin I elected Easiest way to prove that $2^{\aleph_0} = c$ as the target of both the original request and the other related one you found.

Thanks for locating them!