Convergent non-monotone sequences

[Easty]

Homework Statement

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

Homework Equations

The Attempt at a Solution

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

As for part b) I am 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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[tiny-tim]

Hi Easty!

For part a) i get C=[0,3]. Is this correct? I am not sure as to wether 0 and 3 are contained in the set though.

Well, by definition of union, 0 is only in C if it's in one of the C_ns … is it?

As for part b) I am 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.

hmm … seems a daft question …

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

 

[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

 

[tiny-tim]

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

Yup!

As for part b would this sequence work:
(1/n*sin(n)).
It should converge by the absolute convergence theorem i think.

Yes, almost any daft sequence works!