For each , define = { + }. Let = { } Note: Ax stands for A subscript x

a) Prove that is countable for every .

b) Prove that is uncountable

Given in question: You may use without proof the fact that a set is countable if and only if there is a sequence

0, 1, 2, ... in which every element of appears.

My proof for (a):

1) Consider the sequence b0, b1, b2... defined by setting b2i = x - i and b(2i +1) = x + i + 1 for all i member of k.

2) Then the sequence {b0, b1, b2 ...} contains every element of Ax. Thus, Ax is countable.

Im stuck on (b).

Please help and see if my proof of part (a) make sense. And help for part (b) as well!