# Convergent non-monotone sequences

1. Apr 30, 2009

### Easty

1. The problem statement, all variables and given/known data

Let C=$$\bigcup$$n=1$$\infty$$Cn where Cn=[1/n,3-(1/n)]
a) Find C in its simplest form.
b)Give a non-monotone sequence in C converging to 0.

2. Relevant equations

3. The attempt at a solution
For part a) i get C=[0,3]. Is this correct? im not sure as to wether 0 and 3 are contained in the set though. Should it be C=(0, 3)?

As for part b) im not really sure here. I thought one such sequence might be
(1, 2, 1/2, 1/3, 1/4, 1/5,...). So the tail converges to zero and the first 2 terms mean it is non-monotone.
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2. Apr 30, 2009

### tiny-tim

Hi Easty!
Well, by definition of union, 0 is only in C if it's in one of the Cns … is it?
hmm … seems a daft question

but I suspect they want it to be non-monotone wherever you start.

3. Apr 30, 2009

### Easty

Ok then so it must be C=(0, 3).

As for part b would this sequence work:
(1/n*sin(n)).
It should converge by the absolute convergence theorem i think.
I'd appreciate any comments or criticisms.
thanks

4. May 1, 2009

### tiny-tim

Yup!
Yes, almost any daft sequence works!