# Rudin's Principles Theorem 1.11 (supremum, infimum)

• ramleren
In summary: And the theorem is saying that if the given set has the property that every non empty subset with an upper bound has a least upper bound, then the set of lower bounds has the property that every non empty subset with a lower bound has a greatest lower bound.In summary, the conversation discusses the concept of the least-upper-bound property and its relationship to the greatest-lower-bound property. The theorem 1.11 from Rudin's analysis book for undergraduates is mentioned and explained, stating that a set with the least-upper-bound property also has the greatest-lower-bound property. The conversation also clarifies the difference between upper and lower bounds and provides an example of a subset that does not have the least-upper-bound property.

#### ramleren

Mod note: Edited by removing [ sup ] tags.
To the OP: Please don't fiddle with font tags, especially the SUP tag, which renders what you write in very small text (superscript).

Hello everyone

I have just started studying mathematics at university this summer and I have decided to supplement my scheduled classes and accompanied books by working through Rudin's analysis book for undergraduates. I am not completely certain, though, that I understand theorem 1.11 found on page five and I hope for some of you to help me a little.

As I understand, a set, say S, has the least-upper-bound property if given any non empty subset of S, say E, which is also bounded above its supremum must lie in S.
So since T={x is in Q: 2<x^2} is a subset of Q and the inf(T)=sqrt(2) is not in Q itself Q does not have the least-upper-bound property, right?

So in my own words theorem 1.11 says something like:

Let S be any set which has the least-upper-bound property and let B be a non empty subset of S which is bounded below. Since it is bounded below there must exists another set whose elements are all lower bounds of B and which consists and all possible lower bounds of B; although the set could theoretically consist of just one element. Call this set L. Conversely, every element of B must function as a upper bound of L. Because S has the least-upper-bound property every subset of S which is not empty and has an upper bound must also have an supremum, I.e. sup(L) exists. Call it £. But because B is defined as consisting of all upper bounds of L we know that £ must lie in B. Also, for every x>£, x must be in B and B only. If we negate it we get x<£ or x=£ and equivalently the opposite of x is only in B, namely x could be outside B. But since B is arbitrary the only way this be certain is for x to be in L. Hence £ is in L. But because every lower bound of B is in L, it's greatest lower bound must also be in L and the largest element in L is £ so it follows sup(L)=inf(B).

I am truly sorry to use that much space, but I cannot formulate myself any better at this point.

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ramleren said:
So in my own words theorem 1.11 says something like:

If you are asking for an evaluation of your own words, if would help if you quoted the wording of the theorem from the book.

ramleren said:
Hello everyone

I have just started studying mathematics at university this summer and I have decided to supplement my scheduled classes and accompanied books by working through Rudin's analysis book for undergraduates. I am not completely certain, though, that I understand theorem 1.11 found on page five and I hope for some of you to help me a little.

As I understand, a set, say S, has the least-upper-bound property if given any non empty subset of S, say E, which is also bounded above its supremum must lie in S.
So since T={x is in Q: 2<x^2} is a subset of Q and the inf(T)=sqrt(2) is not in Q itself Q does not have the least-upper-bound property, right?
You seem to be confusing "upper bound" and "lower bound". Since sqrt(2) is a lower bound for T, it has nothing to do with upper bounds on subsets of T.

So in my own words theorem 1.11 says something like:

Let S be any set which has the least-upper-bound property and let B be a non empty subset of S which is bounded below. Since it is bounded below there must exists another set whose elements are all lower bounds of B and which consists and all possible lower bounds of B; although the set could theoretically consist of just one element. Call this set L. Conversely, every element of B must function as a upper bound of L. Because S has the least-upper-bound property every subset of S which is not empty and has an upper bound must also have an supremum, I.e. sup(L) exists. Call it £. But because B is defined as consisting of all upper bounds of L we know that £ must lie in B. Also, for every x>£, x must be in B and B only. If we negate it we get x<£ or x=£ and equivalently the opposite of x is only in B, namely x could be outside B. But since B is arbitrary the only way this be certain is for x to be in L. Hence £ is in L. But because every lower bound of B is in L, it's greatest lower bound must also be in L and the largest element in L is £ so it follows sup(L)=inf(B).

I am truly sorry to use that much space, but I cannot formulate myself any better at this point.

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Hard to be sure, but I think he is proving that the LUB property implies the INF property. I.e. he assumes that every non empty set which has an upper bound has a least upper bound, and he proves that every non empty set with a lower bound has a greatest lower bound, by showing that the greatest lower bound of the given set actually equals the least upper bound of the set of lower bounds.

As Halls says, we are not concerned here with upper bounds of subsets of the given set, but with upper bounds of a related set, namely of the set of lower bounds of the given set.

## 1. What is Rudin's Principles Theorem 1.11?

Rudin's Principles Theorem 1.11, also known as the Supremum and Infimum Theorem, is a fundamental result in real analysis that states that every non-empty set of real numbers that is bounded above (or below) has a least upper bound (or greatest lower bound).

## 2. How is Rudin's Principles Theorem 1.11 useful in mathematics?

This theorem is useful in proving many important results in mathematics, including the existence of limits of functions, the convergence of sequences and series, and the continuity of functions. It also provides a basis for the definition of the real numbers and the completeness of the real number system.

## 3. What is the difference between a supremum and an infimum?

The supremum of a set is the smallest number that is greater than or equal to all the numbers in the set. In contrast, the infimum of a set is the largest number that is less than or equal to all the numbers in the set. In other words, the supremum is the least upper bound, while the infimum is the greatest lower bound.

## 4. Can you give an example of how Rudin's Principles Theorem 1.11 is applied in a real-life situation?

Sure, one example is in finance. The prices of stocks and commodities are bounded by certain limits, and Rudin's Principles Theorem 1.11 can be used to determine the maximum or minimum price that a stock or commodity can reach. This can be helpful in making investment decisions.

## 5. Is Rudin's Principles Theorem 1.11 a difficult concept to understand?

It can be a challenging concept to grasp at first, especially for those new to real analysis. However, with practice and a solid understanding of basic mathematical concepts such as sets, limits, and bounds, one can understand and apply this theorem effectively.