Question

give eg..
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 inc

Answer 1

1. (-1)^n

Answer 2

\((-1)^n\) is a bounded sequence...

You can use a different kind of alternating sequence that's unbounded, but still contains a convergent subsequence, such as
\[\left\{1,\pi,2,\pi,3,\pi,...\right\}\]
The particular subsequence of interest is \(\left\{\pi,\pi,\pi,...\right\}\), which obviously converges to \(\pi\).

Answer 3

For (5), you can consider the following sequences:
\[\{a_n\}=\{-1,1,-1,1,...\}\\
\{b_n\}=\{2,0,2,0\}\]
Then you have \(a_1+b_1=a_2+b_2=a_3+b_3=\cdots=1\), so \(\{c_n\}=\{1,1,1,1,...\}\), yet neither \(a_n\) nor \(b_n\) converge.

Answer 4

For (2), I would think the natural numbers \(\{1,2,3,...\}\) would suffice, but not totally sure.