Proof about sequence properties

[cragar]

Homework Statement

Use the Monotone convergence theorem to give a proof of the Nested interval property.

Homework Equations

Monotone convergence theorem: If a sequence is increasing or decreasing and bounded then it converges.
Nested Interval property: If we have a closed interval [a,b] and we keep making intervals inside this and they keep getting smaller the union of all these intervals is non-empty and contains one element.

The Attempt at a Solution

If we started at the left endpoint of some closed interval and we had a monotonically increasing sequence and it continued on the to right with equally spaced steps, and we had a decreasing sequence that started from the right endpoint, eventually these 2 sequences will be heading towards each other and eventually reach the same common point. I think I need to be careful about how I pick the spacing between the terms in the sequence. Am I headed in the right direction with this.

 

[micromass]

You're doing ok. You indeed have to consider the sequence of end-points. This proves that there is an element in all the nested intervals.

 

[cragar]

Ok so let's say we have a closed interval [a,b] and we have a sequence that is increasing and it starts at a and steadily goes up to b in equal steps. Now let's take this same sequence and start it at b and run it backwards b going down to a. Now if we start at a and go up to b, and each time we move from a we also move from b so we have elements that are common to all intervals because they are enclosed inside each other because the sequence is steadily increasing and decreasing. Now we have to consider where the 2 endpoints are getting close to each other, these 2 sequences will eventually reach the same point because we constructed it from the same sequence. So there will be a point that is common to all intervals. let's assume that there wasn't a point that was common to all intervals. This would imply that there was a point where the 2 sequences jumped passed each other, but this couldn't happen because the 2 sequences would have to reach the same point because we constructed them this way to make that happen.