The second step is to show that there is a subset K of R such that there is no surjection (and thus no bijection) from N to K. The proof of (i) is the same as the pro...The second step is to show that there is a subset K of R such that there is no surjection (and thus no bijection) from N to K. The proof of (i) is the same as the proof that T is uncountable in the proof of Theorem 1.20. Keep a counter c∈N that marks the point (0,0) with a “1”. Up the value of the counter by 1 whenever you hit a point of Z2.
This means that every term of the new sequence is different from the corresponding term of the diagonal sequence. (This idea of choosing a sequence that is completely different from the diagonal is ca...This means that every term of the new sequence is different from the corresponding term of the diagonal sequence. (This idea of choosing a sequence that is completely different from the diagonal is called Cantor diagonalization, because it was invented by the mathematician Georg Cantor.) Also, to avoid problems coming from the fact that .999⋯=1.000⋯, you should not use the digits 0 and 9.
