Bisecting indefinitely a real interval

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In summary, if you want to touch a number in the interval [0, 5], you must bisect [0, 5] infinite times.
  • #1
Castilla
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(A, B) is an interval in the real line. If I take the middle point (B-A)/2, and then take the middle point of each of the two subintervals of (A,B), and then take the middle point of each of the four subintervals of (A, B) and I go on this indefinitely, can I be sure that sooner or later I will touch any point of (A, B)?

(I know that there are infinitely many irrationals in (A, B) but I am carrying this process "to the limit").

Maybe dumb question but I need your support.

Thanks.
 
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  • #2
I'm not sure if you mean it like that but when you write the interval (a,b), that is an open interval meaning the end points are excluded (a and b itself). Of course, you'll never reach a and b this way.
The interval with the endpoints included, which is the closed interval, is usually written [a,b].
 
  • #3
What do you precisely mean by 'touch' and 'go on indefinintely'?

I think the answer you want is 'yes, you can get arbitrarily close to any number in the interval by considering the points in this construction: given we've done the split n times then two neighbouring points we construct are 2^n apart. Any number in the whole interval lies in one of these subintervals so there is some point we construct that is less than 2^{n+1} from it, and we can make 2^{n+1} as small as we choose.
 
  • #4
bisecting interval

Why can't I put a space between paragraphs?

TD, I mean [A, B].
Thanks, Matt. I think your answer is enough.
The reason of my question was this. (Excuse but I can't use latex. Any time i put the symbols i got: "Latex graphic is being generated. Reload this page in a moment", and never go beyond. So I have to put all in words).
With "P" I mean any partition of [a, b] and with "P_n" I mean a partition of [a,b] constructed bisecting n times the interval. So any P_n is a P but not viceversa.
In one proof Apostol implies that, as the Riemann integral of the function "f" over [a, b] is equal to the supremum of {lower sums of "f" over any partition P of [a, b]}, then it is also equal to the supremum of {lower sums of "f" over any partition P_n of [a,b]}.
My doubt was: The set {lower sums of "f" over a partition P_n of [a, b]} is a proper subset of the set {lower sums of "f" over a partition P of [a, b]}, so it is not automatically valid to say that the supremum of the bigger set is also the supremum of the smaller one. There could be an element in the bigger set that: 1) is less than the supremum of the bigger set, and 2) it is not on the smaller set. So this element would be an upper bound of the smaller set and less than the supremum of the bigger set. So the supremum of the bigger set would not be the least upper bound of the smaller set.

But your answer tells me that if I take any partition P I can aproximate beoynd all limit it by some partition P_n so that element I mentioned can not exist and therefore the supremum of the bigger set is also supremum of the smaller one.
Excuse the poor redaction. I am not so fluent in english.
 
  • #5
This problem can be simplified.

Imagine, you want to get to the number 2. You start by adding the half of 2, 1. 0+1=1, *dUh*. Then, you add half of 1, 0.5. You get 1.5. Next, you get 1.75, 1.875, etc, etc. If you go on infinitely, you will eventually get to 2.
 
  • #6
Ah, the criterion of integrability. Yes, it is slightly surprising that its true: if you can find one sequnce of partitions whose 'length' tends to zero and for which the upper and lower riemann sums converge and agree then the function is integrable (ie works for all partitions when the length tends to zero).

I think the 'big' reason for this is that since the real numbers are defined in terms of sequences you can prove things in terms of sequences. Some spaces are far too complicated for you to determine behaviour based upon merely sequences (things indexed by N), and you need to pass to things called ultra-filters.
 
  • #7
Thanks, Matt.

Castilla.
 
  • #8
Thanks, nagsjunk, your example is fine though it is not exactly the case.

Suppose you have interval [0, 5]. You want to "touch" 2. You can not bisect [0, 2] and then bisect [1, 2] and then bisect [1.5, 2], that way is not allowed. The only "road" you are allowed to take is to bisect [0, 5] and then to bisect [0, 2.5], and then to bisect [1.25, 2.5], and then to bisect etcétera...
 

1. What does it mean to bisect indefinitely a real interval?

Bisecting indefinitely a real interval means to repeatedly divide a given interval into two equal parts, with each division resulting in two smaller intervals of equal length. This process can continue infinitely, resulting in an infinite number of smaller intervals within the original interval.

2. Why is bisecting indefinitely a real interval important in science?

Bisecting indefinitely a real interval is important in science because it allows for the precise measurement and analysis of continuous quantities. By dividing a real interval into smaller and smaller parts, scientists can obtain more accurate and detailed data, which can lead to a better understanding of natural phenomena and processes.

3. What is the mathematical concept behind bisecting indefinitely a real interval?

The mathematical concept behind bisecting indefinitely a real interval is known as the "limit" in calculus. As the interval is divided into smaller and smaller parts, the limit represents the value that the interval approaches but never reaches. This concept is fundamental in understanding the behavior of continuous functions.

4. Can bisecting indefinitely a real interval be applied to any type of interval?

Yes, bisecting indefinitely a real interval can be applied to any type of interval, whether it be a length, time, temperature, or any other continuous quantity. This method can be used in various fields of science and mathematics, such as physics, biology, and economics.

5. What are the potential limitations of bisecting indefinitely a real interval?

One potential limitation of bisecting indefinitely a real interval is that it requires an infinite amount of time and resources to reach the limit. In practical applications, scientists may need to stop at a certain point due to limitations in technology or funding. Additionally, the accuracy of the results may also be affected by factors such as measurement errors or assumptions made during the process.

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