show that if set B is symmetric, that is B=-B, then $\bigcup_{n=1}^\infty B_n$ is also symmetric

I’ve been trying to proof the following:

If the set $B\subset R$ is symmetric, that is B=-B

Let $B_n=(-n,n)$ with $n \in \mathbb{R^+}$

$\bigcup_{n=1}^\infty B_n$ is also symmetric

I’ve tried to work with the definitions of unions with “there exists an element x in the union” and then just put -x is also is in the union but it seems like pulling out of thin air. What would be a proper proof technique?