I have this exercise where we let $X$ be a topological space and $A\_n\subseteq X$ $n\in\mathbb{N}$ connected subspaces. Suppose $A\subseteq X$ is another connected subspace satisfying $A\cap A\_n\neq \emptyset$ for every $n\in\mathbb{N}$. Show that $A\cup \bigcup\_{n\in\mathbb{N}}\subseteq X$ is connected.

I am thinking that I let B and C be open disjoint subsets. And I assume

$$A\cup \bigcup\_{n\in\mathbb{N}}A\_n=B\cup C$$

Then I want to show that $B=A\cup \bigcup\_{n\in\mathbb{N}}A\_n$ and $C=\emptyset$, but I am confused about how to do so.

Or the other way I am thinking I can do this is by contraction, so I'll assume that $A\cup \bigcup\_{n\in\mathbb{N}}A\_n$ is not connected, hence we'll have non-empty disjoint open subsets B and C such that $A\cup \bigcup\_{n\in\mathbb{N}}A\_n=B\cup C$. From this I am thinking I want to obtain either that A is not connected, there's exists a $n\in\mathbb{N}$ for which $A\_n$ is not connected, or $A\cap A\_n =\emptyset$, but I am confused about how to do so.