Reading notes on the Poincare Recurrence Theorem and I am a bit stuck with Theorem 1.
For a measure preserving dynamical system $T: X \to X$ we have $\mu(T^{-1}E) = \mu(E)$. Why does it "easily follow" that:
$$ \sum_{n=1}^N \sum_{m=1}^N \mu\bigg(T^n E \cap T^m E\bigg) \leq N^2 \bigg( \limsup_{n \to \infty} \mu(E \cap T^n E) + o(1) \bigg) $$
In fact, let's forget entirely we are dealing with dynamical systems... why is it easily true that for positive numbers $x_k \geq 0$
$$ \sum_{n=1}^N \sum_{m=1}^N x_{m-n} \leq N^2 \bigg( \limsup_{n \to \infty} x_n + o(1) \bigg) $$
It is not so bad, it's just that everything should be smaller than the limsup: $\displaystyle \boxed{x_k \leq \limsup_{n \to \infty} x_n }$
Why is the $o(1)$ still necessary?