His second "proof" of uncountability is very poorly explained. Strictly speaking it is false, since all his reasoning applies equally to the rationals. What he shows is that a countable set would have zero measure. You have to also show that (say) the real interval [0,1] has measure 1 (or at least positive measure) to get a contradiction. That requires some more work. You have to be using some property of the reals in order to prove uncountability, as of course the Cantor diagonal argument does.