I was given: $$A_{n,m}=\left\{x \in \mathbb{R} \ \middle|\ \frac{n-1}{m+1}\leq x < n + m\right\}, \quad n, m \in \mathbb{N}$$
And tasked with finding sets:
$$\bigcup_n \bigcap_m A_{n,m}\quad\quad\bigcap_n \bigcup_m A_{n,m}$$
But I don't know how to approach chaining infinite operators.
Does the $\bigcup_n \bigcap_m A_{n,m}$ mean a set of elements that satisfy condition for every $m$ and any $n$ and $\bigcap_n \bigcup_m A_{n,m}$ is a set of $x$s that satisfy the condition for any $m$ and every $n$?
Similarly, I can't interpret multiple infinite operators in problem below:
Let $T = \bigcup_{s\in S}T_s$ and $\mathcal{K}$ be a family of all subsets of $T$ that have at least one common element with each of $\;T_s$ sets.
Prove that if $\mathcal{K} \neq \emptyset$:
$$\bigcup_{s\in S}\bigcap_{t\in T_s}A_t=\bigcap_{Y\in \mathcal{K}}\bigcup_{t\in Y}A_t$$