- #1
EdMel
- 13
- 0
Please let me know if this should be in the Homework section.
Part of Rudin 2.41 says that for a set E in ℝk... "If E is not bounded, then E contains points xn with |xn|>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 points xn is infinite and clearly has no limit point in ℝk". I can understand that S is infinite but to me S clearly does have limit points in ℝk. 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
Part of Rudin 2.41 says that for a set E in ℝk... "If E is not bounded, then E contains points xn with |xn|>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 points xn is infinite and clearly has no limit point in ℝk". I can understand that S is infinite but to me S clearly does have limit points in ℝk. 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