Give examples of the following: 1.an unbounded sequence that has a convergent subsequence . 2.an unbounded sequence that has no convergent subsequence . 3.a null sequence (an) such that the series ¡æ an does not converge . 4.a sequence which is not a cauchy sequence but has the property that for every ¦å > 0 and every n > 0 there exists n > n and m > 2n such that mod(an ¨c am) ¡ü ¦å . 5.two sequences (an) and (bn) such that the sequence (cn) defined by cn = an + bn converges to 1 but neither (an) nor (bn) converge. 6.a sequence (an) that tends to +ve infinity but is neither increasing nor eventually increasing

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)nlog⁡n, consider any real x. For n≥e|x|+1n≥e|x|+1, |(−1)nlogn−x|≥1|(−1)nlog⁡n−x|≥1, so no subsequence of the sequence converges to x.