In Cantor's diagonalization we construct a number x between 0 and 1 that's not on the supposed list of real numbers between 0 and 1. Recall, to construct x we make x's  ith digit (after the decimal point) equal to 1 if the corresponding digit of the ith number on the list is even and we make x's  ith digit 0 otherwise.

a) Suppose the list happens to start with the numbers 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 20.2/3, pi/4, sqrt(3) /2, 1/e and 1/13. What are the first eleven digits of x, after then decimal?
b) Explain why x cannot be on the list, even though it's a number between 0 and 1. Why does this lead to a contradiction?
c) Cantor's proof shows that the real numbers between 0 and 1 aren't countable. Suppose we try to use the same argument to show that the rational numbers between 0 and 1 aren't countable? (Of course we know that they are....) Where does the proof break down?