Where does the less than or equal come from in the proof of density of Q in R?

  • Thread starter completenoob
  • Start date
In summary, the conversation is about a person who is starting to study analysis on their own and is having trouble understanding a proof in Rudin's Principles of Mathematical Analysis. They ask for clarification on the last line of the proof, and other users suggest alternative texts and resources for self-study.
  • #1
completenoob
26
0
Hello,

I have decided to study analysis on my own and am starting with principles of mathematical analysis by rudin.
I am having trouble understanding pg. 9 on the density of Q in R, part b.

It states:
If [itex]x \in R, y \in R [/itex] and [itex] x<y [/itex] the there exists a [itex] p \in Q [/itex] such that [itex]x < p < y[/itex]
Proof:
Since [itex] x<y[/itex], we have [itex] y-x>0[/itex] and the Archemedian Property furnishes a positive integer n such that:
[itex]n(y-x)>1[/itex]
Applying the AP again, to obtain positive integers m1 & m2 such that [itex]m1>nx[/itex], [itex]m2>-nx[/itex]
Then: [itex]-m2<nx<m1 [/itex]
Hence there is an integer m such that [itex]m-1 \le nx<m [/itex]...

Can someone explain to me the last line? Where does this less then or equal come from?
 
Physics news on Phys.org
  • #2
:cry:
 
  • #4
  • #5
MMmm...For being my first analysis book Principles of Mathematical Analysis by Rudin might be a little too hard. Should I study Apostol's Principles of Mathematical Analysis first? Then go back to Rudin's text?
 
  • #6
Probably depends how much you already know.
 
  • #7
Calc 1-3. No analysis experience.
 
  • #8
Yah i need a text for self study. I have no experience with proofs.
 
  • #9

Related to Where does the less than or equal come from in the proof of density of Q in R?

1. Where does the less than or equal come from in the proof of density of Q in R?

The less than or equal sign, denoted as <=, is a mathematical symbol that represents the relationship between two numbers. It is used to indicate that the first number is less than or equal to the second number. In the proof of density of Q in R, the less than or equal sign is used to show that every real number in R can be approximated by a rational number in Q, or in other words, that Q is dense in R.

2. How is the less than or equal sign used in the proof of density of Q in R?

In the proof of density of Q in R, the less than or equal sign is used to show that for any real number x in R, there exists a rational number q in Q such that x <= q. This means that q is either equal to x or is slightly larger than x, but still remains less than or equal to x. This is a crucial step in proving that Q is dense in R.

3. Why is the less than or equal sign important in the proof of density of Q in R?

The less than or equal sign is important in the proof of density of Q in R because it allows us to show that there is no gap between the rational numbers in Q and the real numbers in R. This means that for any real number x in R, we can always find a rational number q in Q that is either equal to x or very close to x. This is what makes Q dense in R.

4. Can the less than or equal sign be replaced with just a less than sign in the proof of density of Q in R?

No, the less than or equal sign cannot be replaced with just a less than sign in the proof of density of Q in R. This is because the less than or equal sign includes the possibility of equality, which is necessary for showing that there is no gap between Q and R. If we only used a less than sign, it would imply that there is a gap between Q and R, which contradicts the proof of density.

5. Is the less than or equal sign unique to the proof of density of Q in R?

No, the less than or equal sign is a commonly used mathematical symbol in various proofs and equations. It is not unique to the proof of density of Q in R, but it plays a crucial role in showing the relationship between Q and R and proving the density of Q in R. It is also used in many other mathematical concepts, such as inequalities and ordered sets.

Similar threads

There are rationals between any two reals? - Rudin 1.20
    • Calculus
Replies
6
Views
4K
Why do we need an m between -m2 and m1 to prove Theorem 1.20?
    • Calculus
Replies
4
Views
3K
Proving Completeness property of ##\mathbb{R}## using Dedekind cuts
    • Calculus and Beyond Homework Help
Replies
20
Views
2K
Bounded non-decreasing sequence is convergent
    • Calculus and Beyond Homework Help
Replies
5
Views
1K
Analysis Understanding how they thought of math proofs in the first place
    • Science and Math Textbooks
Replies
10
Views
2K
A Monte Carlo Wang-Landau_antiferromagnetic perovskite
    • Atomic and Condensed Matter
Replies
3
Views
902
Why must q be the least element for (q+1)a to be greater than b?
    • Calculus and Beyond Homework Help
Replies
2
Views
939
MHB Proof that the integral is independent from the path from A to B
    • Calculus
Replies
4
Views
2K
Proof of the Archimedean Property
    • General Math
Replies
9
Views
2K
Q is dense in R question about proof
    • Topology and Analysis
Replies
5
Views
16K

Hot Threads

Recent Insights

Back
Top