Okay, so I'm being asked to prove that there exists and such that is (Lebesgue) integrable in where is dense in . My problem is I'm pretty sure we need some coefficients to control the series, here's my reasoning:
Let and such that . Then since is dense there are an infinite number of elements of the sequence in and in . Now if the series obviously diverges for all so assume first that then there is such that (taking the subsequence to be in ) and so so the series diverges at . Using an analogous argument for (taking the subsequence to be in and noting that the exponent is negative and thus reverses inequalities) we see that the series diverges. Since was arbitrary we get that which is nonsense, since an integrable function attains at most in a null set.
Does this look right? I'm not too sure myself since I'm not overly comfortable with my series knowledge, but it feels right (since we don't really know if for all which actually might never happen).
Any thoughts and comments are appreciated.