Let $n \geq 1$ be a postive integer, and let $X_{1}$,...,$X_{n}$ be closed subsets of R. Show that $ X_{1}$$\cup$$ X_{2}$$\cup$...$\cup$$X_{n}$ is also closed.

My attempt

I know that a subset $E \subseteq R$ is said to be closed if $ \bar{E} = E$, in other words that E contains all of its adherent points. So I believe I need to show that the union of all these sets needs to contain all of the adherent points of each individual set.


Every convergent sequence that lives in the union of closed subsets will converge to a point in the union, hence the union is a closed set.

Here I am assuming that all your subsets are proper subsets of $\mathbb R$, so that no set is $\mathbb R$ itself. (Remember that $\mathbb R$ is both open and closed.)


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