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Converse of Bolzano-Weierstrass theorem
Clash Royale CLAN TAG #URR8PPP up vote 1 down vote favorite Bolzano-Weierstrass theorem states that every bounded sequence has a limit point. But, the converse is not true. That is, there are some unbounded sequences which have a limit point. In my course book, I found an example for this claim, but it doesn't make sense. Here's the example give in the book: The set: 1, 2, 1, 4, 1, 6, ... is unbounded, but has a limit point of 1 . I can't understand how this set has a limit point as 1 . According to the book definition of limit point, 'x' is the limit point of a sequence, if every neighborhood of 'x' has infinitely many elements of the sequence. If I apply it here, then I get only infinity as the limit point. Am I missing something? real-analysis sequences-and-series limits share | cite | improve this question edited 36 mins ago Eric Wofsey 169k 12 197 314 asked 49 mins ago Sankalp 108 4 New contributor