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Part of Rudin 2.41 says that for a set E in ℝ^{k}... "If E is not bounded, then E contains pointsxwith |_{n}x|>_{n}n(n= 1, 2, 3,...)." I can understand the argument this far. I do not get the next sentence "The set S consisting of these pointsxis infinite and clearly has no limit point in ℝ_{n}^{k}". I can understand that S is infinite but to me S clearly does have limit points in ℝ^{k}. For example, take the real line, n = 1, S to be the set of points greater than 1 and ε>0. Every neighborhood of p = 2 with radius ε would contain a point in S making p a limit point. This contradicts what Rudin states as 'clearly'. What am I missing here?

Thanks,

Ed

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# Rudin 2.41, why does an unbounded infinite set have no limit points?

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