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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

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.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
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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