# Thread: Nested Intervals and IVT

1. ## Nested Intervals and IVT

I could really use some help. Not really sure how to get started.

Suppose $\displaystyle f(a,b)< 0$ and $\displaystyle f(c,d)>0$. Construct nested intervals $\displaystyle [a_{n},b_{n}]$ and $\displaystyle [c_{n},d_{n}]$ such that $\displaystyle f(a_{n},c_{n})\leq 0$ and $\displaystyle f(b_{n},d_{n})>0$. Then show $\displaystyle f(x_{0},y_{0})=0$ if $\displaystyle \bigcap _{n=1}^\infty [a_{n},b_{n}]=\{x_{0}\}$ and $\displaystyle \bigcap _{n=1}^\infty [c_{n},d_{n}]=\{y_{0}\}$.

2. Originally Posted by zebra2147
I could really use some help. Not really sure how to get started.

Suppose $\displaystyle f(a,b)< 0$ and $\displaystyle f(c,d)>0$. Construct nested intervals $\displaystyle [a_{n},b_{n}]$ and $\displaystyle [c_{n},d_{n}]$ such that $\displaystyle f(a_{n},c_{n})\leq 0$ and $\displaystyle f(b_{n},d_{n})>0$. Then show $\displaystyle f(x_{0},y_{0})=0$ if $\displaystyle \bigcap _{n=1}^\infty [a_{n},b_{n}]=\{x_{0}\}$ and $\displaystyle \bigcap _{n=1}^\infty [c_{n},d_{n}]=\{y_{0}\}$.
What is this $\displaystyle f$ just some continuous map $\displaystyle f:[0,1]\times[0,1]\to\mathbb{R}$? What are $\displaystyle a,b,c,d$? I think you need to type up the full question.

3. $\displaystyle f$ is a continuous function on some interval $\displaystyle I$ where $\displaystyle a,b,c,d\in I$.

4. Originally Posted by zebra2147
$\displaystyle f$ is a continuous function on some interval $\displaystyle I$ where $\displaystyle a,b,c,d\in I$.
Ok? Then what does $\displaystyle f(x,y)$ mean? Unless you meant $\displaystyle f\left((a,b)\right)$ but then the inequality signs used are meaningless.

5. There was a typo to this problem initially. It should read...
Suppose $\displaystyle f(a,c)<0$ and $\displaystyle f(b,d)>0$.

I'm guessing we need to use the Intermediate Value Theorem to show that if there exists $\displaystyle f(x_0,y_0)=0$ then $\displaystyle f(a,c)<f(x_0,y_0)<f(b,d)$. And then somehow use this to show that $\displaystyle \bigcap _{n=1}^\infty [a_{n},b_{n}]=\{x_{0}\}$ and $\displaystyle \bigcap _{n=1}^\infty [c_{n},d_{n}]=\{y_{0}\}$??

6. Originally Posted by zebra2147
There was a typo to this problem initially. It should read...
Suppose $\displaystyle f(a,c)<0$ and $\displaystyle f(b,d)>0$.

I'm guessing we need to use the Intermediate Value Theorem to show that if there exists $\displaystyle f(x_0,y_0)=0$ then $\displaystyle f(a,c)<f(x_0,y_0)<f(b,d)$. And then somehow use this to show that $\displaystyle \bigcap _{n=1}^\infty [a_{n},b_{n}]=\{x_{0}\}$ and $\displaystyle \bigcap _{n=1}^\infty [c_{n},d_{n}]=\{y_{0}\}$??
You have yet to answer my question. What does $\displaystyle f(a,c)<0$ mean?

7. Well, my professor never told us but judging by the position of the proof in my notes I think we are trying to help prove the following theorem:

If $\displaystyle I$ and $\displaystyle J$ are intervals and $\displaystyle f:I\times J\rightarrow \mathbb{R}$ is continuous, then the range of $\displaystyle f$ is an interval.

So is it possible that $\displaystyle f(a,b)$ is an interval??

8. Originally Posted by zebra2147
Well, my professor never told us but judging by the position of the proof in my notes I think we are trying to help prove the following theorem:

If $\displaystyle I$ and $\displaystyle J$ are intervals and $\displaystyle f:I\times J\rightarrow \mathbb{R}$ is continuous, then the range of $\displaystyle f$ is an interval.

So is it possible that $\displaystyle f(a,b)$ is an interval??
But isn't it more likely given what you just said that you're suppose to consider $\displaystyle (a,b)\in I\times J$?

9. Ok well if we consider $\displaystyle (a,b)\in I\times J$ could you help me get started?

10. Originally Posted by zebra2147
Ok well if we consider $\displaystyle (a,b)\in I\times J$ could you help me get started?
I'm sorry, no. Until I understand what's going on, I can't say anything intelligible.