You might have missed Jacob Plotkin’s comment a week ago about about a reference (by me!) to “Cantor’s back-and-forth proof”:
Cantor did not invent/discover or use this method of proof. That honor belongs independently to Felix Hausdorff and E.V. Huntington.
Jacob gave a reference to the short paper where he spells out the evidence. I’ve now had a chance to read it, and it is interesting and instructive (and after all, it is always better to get our history right rather than wrong). So let me add a link to where you can find the paper!