Diagonalization might be a tricky topic for beginners, but that is also who we teach it to. It is usually in your introduction to mathematical proofs class, and

Diagonalization might be a tricky topic for beginners, but that is also who we teach it to. It is usually in your introduction to mathematical proofs class, and even there it is taught in the first couple of weeks. There is an additional problem with the argument as written, as there are in fact "relations" on the "decimal expansions" of real numbers. for instance .099... = .1000.... Notice that none of the places in this expansion agree with each other so this might indeed be the element you construct from the following list: `.1000... .0100... .0010... ...` You might say, ok well pick a number different from a_i and also different from 0 or 9, which does indeed get you a number not in your list, but it really begs the question of "why doesn't this work in binary?", and is also far less intuitive than factoring the problem into two steps of |N|~=|P(N)|=|R|. This approach is more general anyway.