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Properties of the real numbers, discussing isomorphic subsets, nonempty subsets with a greatest lower bound, least upper bound properties, positive square roots, and other aspects of real numbers.

The set R contains a subset that is isomorphic to the ordered field Q of rational numbers, and hence subsets that are isomorphic to N and Z .

REMARK. The proof of [link] is immediate from part (b) of Exercise 1.7. In view of this theorem, we will simply think of thenatural numbers, the integers, and the rational numbers as subsets of the real numbers.

Having made a definition of the set of real numbers, it is incumbent upon us now to verify that this set R satisfies our intuitive notions about the reals. Indeed, we will show that 2 is an element of R and hence is a real number (as plane geometry indicates it should be),and we will show in later chapters that there are elements of R that agree with our intuition about e and π . Before we can proceed to these tasks, we must establish some special properties of the field R . The first, the next theorem, is simply an analog for lower bounds of the least upper bound condition that comes from the completeness property.

If S is a nonempty subset of R that is bounded below, then there exists a greatest lower bound for S .

Define T to be the set of all real numbers x for which - x S . That is, T is the set - S . We claim first that T is bounded above. Thus, let m be a lower bound for the set S , and let us show that the number - m is an upper bound for T . If x T , then - x S . So, m - x , implying that - m x . Since this is true for all x T , the number - m is an upper bound for T .

Now, by the completeness assumption, T has a least upper bound M 0 . We claim that the number - M 0 is the greatest lower bound for S . To prove this, we must check two things. First, we must show that - M 0 is a lower bound for S . Thus, let y be an element of S . Then - y T , and therefore - y M 0 . Hence, - M 0 y , showing that - M 0 is a lower bound for S .

Finally, we must show that - M 0 is the greatest lower bound for S . Thus, let m be a lower bound for S . We saw above that this implies that - m is an upper bound for T . Hence, because M 0 is the least upper bound for T , we have that - m M 0 , implying that m - M 0 , and this proves that - M 0 is the infimum of the set S .

The following is the most basic and frequently used property of least upper bounds. It is our first glimpse of “ limits.”Though the argument is remarkably short and sweet, it will provide the mechanism for many of our later proofs, so master this one.

Let S be a nonempty subset of R that is bounded above, and Let M 0 denote the least upper bound of S ; i.e., M 0 = sup S . Then, for any positive real number ϵ there exists an element t of S such that t > M 0 - ϵ .

Let ϵ > 0 be given. Since M 0 - ϵ < M 0 , it must be that M 0 - ϵ is not an upper bound for S . ( M 0 is necessarily less than or equal to any other upper bound of S . ) Therefore, there exists an element t S for which t > M 0 - ϵ . This is exactly what the theorem asserts.

  1. Let S be a nonempty subset of R which is bounded below, and let m 0 denote the infimum of S . Prove that, for every positive δ , there exists an element s of S such that s < m 0 + δ . Mimic the proof to [link] .
  2. Let S be any bounded subset of R , and write - S for the set of negatives of the elements of S . Prove that sup ( - S ) = - inf S .
  3. Use part (b) to give an alternate proof of part (a) by using [link] and a minus sign.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Analysis of functions of a single variable. OpenStax CNX. Dec 11, 2010 Download for free at http://cnx.org/content/col11249/1.1
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