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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.

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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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