Thor and Loki play the game: Thor chooses an integer $n_1 \ge 1$ , Loki chooses $n_2 \gt n_1$, Thor chooses $n_3 \gt n_2$ and so on. Let $X$ be such that
$$X = \bigcup_{j\in\mathbb N^*} \left(\left[n_{2j-1},n_{2j}\right) \cap \mathbb Z \right)$$
and $$s= \sum_{n\in X} \frac {1}{n^2}$$
Thor wins if s is rational, and Loki wins if s is irrational. Determine who has got the winning strategy.
Attempt: That's basic measure theory, right? Thor can always limit how far the sum can go, but irrationals are denser. Loki always manages to "reach" another irrational number, which does not happen with rational ones. Am I correct, or do I need more justification?