Question

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

Answer 1

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.

Answer 2

Answer:

you got it

Step-by-step explanation: