Your illustration that the manipulations done in the numberphile video are "tricks" by showing that 2Z = Z seems rather weak. You prove that 2Z = Z by appending the sequences to each other; an operation that is never done in the video and seems like an ill defined operation. Where would you start appending? The operations they do do: adding the elements of shifted sequences (Hilbert's hotel fashion) always provides the same unique result for a sequence in whatever way you do the manipulations, so are quite different imho.

Your illustration that the manipulations done in the numberphile video are "tricks" by showing that 2Z = Z seems rather weak. You prove that 2Z = Z by appending the sequences to each other; an operation that is never done in the video and seems like an ill defined operation. Where would you start appending? The operations they do do: adding the elements of shifted sequences (Hilbert's hotel fashion) always provides the same unique result for a sequence in whatever way you do the manipulations, so are quite different imho.