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I think this proof is easy, but would like someone to check my work since sometimes I miss technicalities on these "easy" proofs.

Let K1, ..., Kp be compact sets in R^n. Show that union( Kj, j = 1 to p) is a compact set in R^n.

Proof.

We show that if K1 and K2 are compact then K1 union K2 is compact. Then

apply this fact finitely many times to conclude the original statement.

I have a theorem: A set E in R^n is compact IFF E is a bounded closed set.

Then if K1 and K2 are compact, they are bounded and closed.

Then the union is closed. We define a bounded set as a set contained in the

open ball B(0, r), where 0 = (0, 0, ..., 0). K1, and K2 bounded implies K1

contained in B(0, r1), and K2 countained in B(0, r2). Then pick r = max(r1,

r2). Then K1 union K2 contained in B(0, r). And hence K1 union K2 is

bounded. Thus K1 union K2 is a compact set.

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# Analysis / compactness proof check

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