- #1
_Andreas
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According to the Bolzano-Weierstrass theorem, a bounded sequence has a convergent subsequence. My problem is with the proof. Either I've got a bad textbook, or my reading comprehension is lacking. This is how it's formulated:
Let x1, x2, ... be a bounded sequence. Let E be the set of all numbers in the sequence. The set E is clearly bounded. The set E is either finite or infinite. If E is finite there exists a number that is repeated an infinite amount of times in the sequence x1, x2, ... . This means that there is a sequence of positive integers n1 < n2 < ... such that xn1 = xn2 ... . Therefore xn1, xn2, ... is a convergent subsequence to xn.
For starters, I need to know what the bolded part means. If the sequence x1, x2, ... is 0,1,2,3,4,0,1,2,3,4,0,1,2,3,4, ... , is E={0,1,2,3,4}, or E={0,1,2,3,4,0,1,2,3,4,0,1,2,3,4, ... }?
Let x1, x2, ... be a bounded sequence. Let E be the set of all numbers in the sequence. The set E is clearly bounded. The set E is either finite or infinite. If E is finite there exists a number that is repeated an infinite amount of times in the sequence x1, x2, ... . This means that there is a sequence of positive integers n1 < n2 < ... such that xn1 = xn2 ... . Therefore xn1, xn2, ... is a convergent subsequence to xn.
For starters, I need to know what the bolded part means. If the sequence x1, x2, ... is 0,1,2,3,4,0,1,2,3,4,0,1,2,3,4, ... , is E={0,1,2,3,4}, or E={0,1,2,3,4,0,1,2,3,4,0,1,2,3,4, ... }?