First of all let us show that Rn is not compact. There are two possible definitions of compactness for metric spaces (Bolzano-Weierstrass sequence property and Borel-Lebesgue open cover property), let us disprove both of them for Rn.
Let us consider a sequence un=⎝⎛n0...0⎠⎞ for n∈N. This sequence does not admit a converging subsequence, as for any subsequence the first coordinate diverges to infinity. Therefore, Rn does not admit the Bolzano-Weierstrass property and thus is not compact (as it is metric).
Let us consider an open cover Un=B(0,n) by the open balls centered at origin with a growing radius. This open cover does not admit a finite subcover, as for any finite subcover we have ⋃n≤NUn⊆B(0,N+1)=Rn. Therefore, Rn does not admit the Borel-Lebesgue property and thus is not compact.
To prove that Cn is not compact, we can either use the former approach or use a simpler argument : if Cn were compact, then Rn⊆Cn being a closed subset of a compact space, would be compact too. However we have just proven the contrary, and thus Cn is not compact.
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