First of all let us show that "\\mathbb{R}^n" 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 "\\mathbb{R}^n".
- Let us consider a sequence "u_n=\\begin{pmatrix} n \\\\0 \\\\ ... \\\\ 0 \\end{pmatrix}" for "n\\in \\mathbb{N}". This sequence does not admit a converging subsequence, as for any subsequence the first coordinate diverges to infinity. Therefore, "\\mathbb{R}^n" does not admit the Bolzano-Weierstrass property and thus is not compact (as it is metric).
- Let us consider an open cover "U_n=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 "\\bigcup_{n\\leq N} U_n \\subseteq B(0,N+1)\\neq \\mathbb{R}^n". Therefore, "\\mathbb{R}^n" does not admit the Borel-Lebesgue property and thus is not compact.
To prove that "\\mathbb{C}^n" is not compact, we can either use the former approach or use a simpler argument : if "\\mathbb{C}^n" were compact, then "\\mathbb{R}^n\\subseteq \\mathbb{C}^n" being a closed subset of a compact space, would be compact too. However we have just proven the contrary, and thus "\\mathbb{C}^n" is not compact.
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