Definition 2. For A, B ? R, let
A + B = {a + b | a ? A and b ? B}.
²In a previous version of this homework, I asked you to prove that Q is G?. This is an
unfortunate typo, because you will be proving that Q is actually not G? next week.
2
7. Keep in mind that the Cantor set C has zero length and contains no
intervals. Prove that C + C = [0, 2].
Hint: It is straightforward to show that C + C ? [0, 2]. For the reverse
inclusion, let s ? [0, 2]. We want to find x, y ? C such that x + y = s. For
n ? N, let $C_n$ denote the nth stage in the construction of C. Prove that for
each n, there are $x_n, y_n ? C_n$ such that $x_n + y_n = s$. Keeping in mind that
the sequences ($x_n$) and ($y_n$) do not necessarily converge, show how they can
nevertheless be used to produce the desired x and y in C satisfying x + y = s.