# Give a counter-example that shows Bolzano-Weirstrass is unvalid in IR2

1. Feb 16, 2013

### tsuwal

1. The problem statement, all variables and given/known data
Give a counter-example that shows Bolzano-Weirstrass is unvalid in IR2.

Intro:
Bolzano-Weirtrass theorem says that if a sequence (IN->IR) is bounded then there exists a convergent sub-sequence. (this is shown using the Cauchy sequence concept, showing that a Cauchy sequence is bounded and using the lemma of monotonic sub-sequences)

However, this is not valid valid in IR2, if a sequence (IN->IR2) is bounded then we can't assure that the exists a convergent sub-sequence.

2. Relevant equations

3. The attempt at a solution
It's not easy since you have the tendency of using a pattern. But I guess
Xn=(Cos(n),Sin(n)) might work...

2. Feb 16, 2013

### Dick

Re: Give a counter-example that shows Bolzano-Weirstrass is unvalid in

Can you explain your notation? What are IN, IR and IR2? Bolzano Weierstrass is valid for sequences in $R^2$.

3. Feb 16, 2013

### tsuwal

Re: Give a counter-example that shows Bolzano-Weirstrass is unvalid in

IR=real number set
IN=natural number set

I'm sorry, I thought Bolzano-Weirtrass was not valid im IR^2, my book was not clear in that part. I've consulted Wikipedia and confirmed that it is valid in IR^2. Sorry for the mistake.

Thanks.