I am having trouble verifying exercise 1 here. Can I get some hints/solutions? (It's not homework, I am just reading up on it for my own interest.)
Theorem 2. (Furstenberg multiple recurrence theorem) Let $k \geq 1$ be an integer, let $(X, {\mathcal X}, \mu, T)$ be a measure-preserving system, and let $E$ be a set of positive measure. Then there exists $r > 0$ such that $E \cap T^{-r} E \cap \ldots \cap T^{-(k-1) r} E$ is non-empty.
Exercise 1. Prove that Theorem 2 is equivalent to the apparently stronger theorem in which “is non-empty” is replaced by “has positive measure”, and “there exists $r > 0$” is replaced by “there exist infinitely many $r > 0$“. $\diamond$