How can Bolzano-Weiertrass theorem be used to prove Nested Interval Property?

[cragar]

Homework Statement

Start with the Bolzano-Weiertrass theorem and use it to construct a proof of the Nested Interval Property.

Homework Equations

Bolzano-Weiertrass: Every bounded sequence contains a convergent sub-sequence
Nested Interval Property: Closed intervals nested inside of each other forever is non-empty.

The Attempt at a Solution

If we start with a bounded sequence on a closed interval and then we make it smaller we have a smaller portion of the sequence and so this smaller part must converge to something and we just keep making the interval smaller and we squeeze it down to a point, the sequence must converge to this point because it is the only point in the sequence.
Can I just start with some interval and slowly make it approach the middle by having it increase from the right and decrease to the left till I just have enclosed one point and make it converge to this point.

 

[Deveno]

to use B-W to prove the Nested Interval Property, you don't want to "start with a sequence". you want to start with an arbitrary family of nested closed intervals.

can you think of a way to create a bounded sequence from such a family?

if so, then you can say: by B-W, we know that...

 

[cragar]

When we start with a family of nested closed intervals, By the B-w, we know that there should be a convergent point among these family of intervals.

 

[Deveno]

you can only apply B-W if you have a bounded sequence. what is your bounded sequence?

 

[cragar]

Do I just say I have some generic sequence A and that it is bounded between some interval.

 

[Fredrik]

Since you're proving the nested intervals thing, you need to start with an arbitrary decreasing sequence of closed intervals. If you're going for a proof that involves sequences of real numbers, you will have to use the sequence of intervals to define a sequence of real numbers.