Respuesta :
ake the sequence: $(0,1,0,2,0,3,0,4,0,5,0,6,\cdots)$It is unbounded and it has a convergent subsequence: $(0,0,0,\cdots)$. The Bolzano-Weierstrass theorem says that any bounded sequence has a subsequence which converges. This does not mean that an unbounded sequence can't have a convergent subsequence. What we can conclude is that any unbounded sequence has *at least* one unbounded subsequence.
1down voteAn unbounded sequence can have a convergent subsequence. An example is an=nan=n for n odd, 00for n even.The correct contrapositive of Bolzano-Weierstrass is: a sequence with no convergent subsequence is unbounded.As for (−1)nlogn(−1)nlogn, consider any real x. For n≥e|x|+1n≥e|x|+1, |(−1)nlogn−x|≥1|(−1)nlogn−x|≥1, so no subsequence of the sequence converges to x.
1down voteAn unbounded sequence can have a convergent subsequence. An example is an=nan=n for n odd, 00for n even.The correct contrapositive of Bolzano-Weierstrass is: a sequence with no convergent subsequence is unbounded.As for (−1)nlogn(−1)nlogn, consider any real x. For n≥e|x|+1n≥e|x|+1, |(−1)nlogn−x|≥1|(−1)nlogn−x|≥1, so no subsequence of the sequence converges to x.