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OK, so I'm going through the proof of a theorem in Munkres:
Theorem 28.2. If X is a metrizable space, then the following statements are equivalent:
(1) X is compact
(2) X is limit point compact
(3) X is sequentially compact
There's an argument in (2) ==> (3) which I don't quite get. I'll quote directly:
Assume X is limit point compact. Given a sequence (xn) of points in X, consider the set A = {xn : n is a positive integer}. If the set is finite, then there is a point x such that x = xn, for infinitely many values of n. In this case, the sequence (xn) has a constant subsequence, and therefore converges trivially.
OK, if the set A is finite, we have a finite sequence. By definition, a subsequence yn of a given sequence is obtained from the original one by taking an increasing sequence of positive integers n1 < n2 < n3 < ... and defining yi = xni.
I don't understand how we obtained a constant infinite sequence from a finite sequence?
Thanks for any help on this one, it's really bugging me.
Theorem 28.2. If X is a metrizable space, then the following statements are equivalent:
(1) X is compact
(2) X is limit point compact
(3) X is sequentially compact
There's an argument in (2) ==> (3) which I don't quite get. I'll quote directly:
Assume X is limit point compact. Given a sequence (xn) of points in X, consider the set A = {xn : n is a positive integer}. If the set is finite, then there is a point x such that x = xn, for infinitely many values of n. In this case, the sequence (xn) has a constant subsequence, and therefore converges trivially.
OK, if the set A is finite, we have a finite sequence. By definition, a subsequence yn of a given sequence is obtained from the original one by taking an increasing sequence of positive integers n1 < n2 < n3 < ... and defining yi = xni.
I don't understand how we obtained a constant infinite sequence from a finite sequence?
Thanks for any help on this one, it's really bugging me.