Let $A_k$ be the the set $A_k=\left\{x\in\left.\mathbb{R}\right|x^2-\left(4k+1\right)x+4k^2+2k<0\right\}$
I need to find a simplified way to rewrite the following group : $\mathbb{R}\setminus\left(\bigcup_{k\in z} A_k\right)$
By simplified I mean - not using arbitrary union or arbitrary intersection
The only equality I could think of is the following:$$\mathbb{R}\setminus\left(\bigcup_{k\in z} A_k\right) = \bigcup_{k\in z}\left[2k+1,2k+2\right]$$
But I still had to use arbitrary union, which I can't.
I would appriciate your assitance with finding the answer and prove the equality.
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