Help with nested intervals in Courant please

[osnarf]

I'm having trouble grasping this concept. This is the part in question:

We require that the length of the interval I_n tends to zero with increasing n; that is, that the length of I_n is less than any preassigned positive number for all sufficiently large n. A set of closed intervals I₁, I₂, I₃, ... each containing the next one and such that the lengths tend to zero will be called a nested sequence of intervals. The point x is uniquely determined by the nested sequence; that is, no other point y can lie in all I_n, since the distance between x and y would exceed the length of I_n once n is sufficiently large. Since here we always choose rational points for the end points of the I_n and since every interval with rational end points is described by two rational numbers, we see that every point x of L, that is, every real number, can be precisely described with the help of infinitely many rational numbers.

Why is the point x uniquely determined by the nested sequence? if i pick two rational numbers, no matter how close together they are, surrounding say the square root of two, shouldn't there always be another two rational numbers that are closer?

What does he mean when he says the distance between x and y would exceed the length of I_n? shouldn't you be able to decrease the lengths of both of them as much as you need to because there is an infinite amount of points?

What does he mean by a preassigned positive number? I'm sure that's the reason why I'm not understanding this... you have to place some sort of cap on how small you can go, right?


He says in the footnotes:

It is important to emphasize for a nested sequence that the intervals I_n are closed. If, for example, I_n denotes the open interval 0 < x < 1/n, then each I_n contains the following one and the lengths of the intervals tend to zero; but there is no x contained in all I_n.

Why does this matter? Again, couldn't you just take the point on L directly in the middle of the two open endpoints and have an x in the middle of the smallest I_n?

Please help this is driving me crazy...

 

[osnarf]

Can anybody help me please?

 

[HallsofIvy]

That last point, that the intervals must be closed, is important because we are talking about an infinite sequence of intervals. Consider the case with I_n= (0, 1/n). That is, the intervals are (0, 1), (0, 1/2), (0, 1/3), (0, 1/4), ...

Every one of those intervals is non-empty. What about their intersection? Since none of the intervals contains a negative number or 0, their intersection contains no negative number and does not contain 0. Let "x" be a positive real number. The 1/x is also a positive real number and by the Archimedian property, there exist a positive integer, N, such that N> 1/x. Since x and N are positive, then, x> 1/N and x is NOT in I_N= (0, 1/N). That shows that no positive number can be in the intersection of those intervals and so the intersection is empty.

You have a series of nested intervals, I₁, I₂, ..., I_n, ... with each interval a subset of the previous one and length(I_n) going to 0. We can write each interval in the form $I_n= [a_n, b_n]$. Because the intervals are nested we have $a_n\le a_m\le b_m\le a_m$ for n< m. That says that the sequence $\{a_n\}$ is an increasing sequence having every $b_k$ as an upper bound. By the "monotone convergence" property, that sequence converges to some limit- call it "a". Of course, that a is also the least upper bound on the set $\{a_m\}$ and, since every b_n is an upper bound, we have $a\le b_n$. That is, $\{b_n\}$ is a decreasing sequence having a as a lower bound. By "monotone convergence" again, that sequence converges to some number, b. Since b is the greatest lower bound for the sequence and a is a lower bound, [itexs]a\le b[/itex]. Every number in [a, b] is less than or equal to every b_n and larger than or equal to every a_n and so in all of the intervals. The intersection of all intervals is [a, b] which is non-empty (if a= b then it contains the single point a- if a< b, then it contains an infinite number of points).

Now suppose a< b- that is suppose the interval [a, b] contains more than one point. Let "x" and "y" be two points in that interval. Notice that we have chosen x and y at this point- they are not "variables" now. They have a specific positive distance between them. Yes, given any N, we could choose x and y so that the distance between x and y is less than b_n- a_{n[/b] but that is not what we are doing here. That would be picking N first, then picking x and y. Here we are picking x and y first. Now, once the two points x and y are fixed, with y not equal to x, the distance between x and y, |x- y| is a positive number. Since the lengths bn- an go to 0, there exist N such that bn- an< |x- y|- and so it is impossible for both x and y to "fit" inside that interval and so cannot be in the intersection of all the intervals. That shows there cannot be two distinct points in the interval [a, b]. We must have a= b and the intersection of those nested, closed, intervals is a single point.}

 

[osnarf]

thank you! i understand now, I'm going to go reread that part in the book now, should make a lot more sense this time around.

 

[osnarf]

Nope. I still don't get it.

If a = b, since a is increasing and b is decreasing and [a,b] is the intersection of all I sub n (sorry I am on my cell phone computer isn't working), that means that and b must be rational, right? since all the endpoints of the intervals that we chose had to be rational. so how do you describe an irrational number then with closed intervals?

sorry for being so thick headeim just not understanding it its been bugging me for weeks now

 

[CRGreathouse]

so how do you describe an irrational number then with closed intervals?

[3, 4]
[3.1, 3.2]
[3.14, 3.15]
[3.141, 3.142]
[3.1415, 3.1416]
[3.14159, 3.1416]
[3.141592, 3.141593]
. . .

 

[disregardthat]

Maybe you should think in terms of convergent sequences. The end points of the intervals are just respectively increasing and decreasing sequences of rational numbers converging to a common irrational number.

 

[osnarf]

That would make sense except he just showed that there is no number in a series of infinite converging intervals so what is the difference if you use closed intervals? The only difference is you are including the end points. so if there was no number contained in all open I sub n then the number contained in all closed I sub n must be one of the end points, right? And we were only allowed to chose rational endpoints.

so my question is how do you enclose an irrational number in all closed I sub n with rational endpoints if you can't enclose an irrational number in all open I sub n with rational endpoints?

 

[CRGreathouse]

That would make sense except he just showed that there is no number in a series of infinite converging intervals

No, what was shows is that there exists a series of converging intervals which have no numbes in their intersection. There are certainly converging intervals (open or closed, take your pick) that contain one or more numbers.

so my question is how do you enclose an irrational number in all closed I sub n with rational endpoints if you can't enclose an irrational number in all open I sub n with rational endpoints?

I gave an example above.