An infinite union of closed sets is a closed set? Question: {$B_n$}$ \in \Bbb R$ is a family of closed sets.
Prove that $\cup _{n=1}^\infty B_n$ is not necessarily a closed set.
What I thought: Using a counterexample: If I say that each $B_i$ is a set of all numbers in range $[i,i+1]$ then I can pick a sequence $a_n \in \cup _{n=1}^\infty B_n$ s.t. $a_n \to \infty$ (because eventually the set includes all positive reals) and since $\infty \notin \Bbb R$ then $\cup _{n=1}^\infty B_n$ is not a closed set.
Is this proof correct?
Thanks

Every subset of $\mathbb R^ n$ is a union of closed sets, namely, the one-point sets consisting of each one of its points.
Yet not all subsets of $\mathbb R^n$ are closed!
