Dense subset of R

A set AA is said to be a dense subset of RR if x,yR,x<y\forall x,y\in R,x<y, aA\exists a\in A, such that x<a<yx<a<y. My this note is to show that the set S={m+n2:m,nZ}S=\left\{ m+n\sqrt { 2 } :m,n\in Z \right\} is dense in RR. For that we observe that SS satisfies the following properties:

i)Additivity: If x,ySx,y\in S, then (x+y)S\left( x+y \right) \in S.

ii)Homogeneity in ZZ: If xSx\in S and kZk\in Z, then kxSkx\in S

Now we observe that nZ\forall n\in Z, 0(n2n2)<10\le \left( n\sqrt { 2 } -\left\lfloor n\sqrt { 2 } \right\rfloor \right) < 1.

Also we note that m,n,m,nZ\forall m,n,m',n'\in Z, m+n2=m+n2m=mandn=nm+n\sqrt { 2 } =m'+n'\sqrt { 2 } \\ \Rightarrow m=m'\quad and\quad n=n'

Now, let sn=(n2n2),nZ{ s }_{ n }=\left( n\sqrt { 2 } -\left\lfloor n\sqrt { 2 } \right\rfloor \right) ,n\in Z Then 0sn<10\le { s }_{ n }< 1 and snS{ s }_{ n }\in S. Also if nnn\neq n', then snsn{ s }_{ n }\neq { s }_{ n' }.Hence {sn:nZ}\left\{ { s }_{ n }:n\in Z \right\} is an infinite subset of S[0,1)S\cap [0,1) .

Now we shall show that for any ε>0\varepsilon >0, sS\exists s\in S such that 0<s<ε0<s<\varepsilon . For that, given any ε>0\varepsilon >0, we partition [0,1)[0,1) into kk equal subintervals, each having size less than ε\varepsilon . Then, since {sn:nZ}\left\{ { s }_{ n }:n\in Z \right\} is an infinite subset of S[0,1)S\cap [0,1) , so at least one subinterval must contain two distinct elements of S[0,1)S\cap [0,1) , say sn,sn{ s }_{ n },{ s }_{ n' }. Without loss of generality, let sn<sn{ s }_{ n }<{ s }_{ n' }. Then obviously 0<snsn<ε0<{ s }_{ n' }-{ s }_{ n }<\varepsilon . Now, from the aforesaid properties i) and ii) of SS, we have that (snsn)=sS\left( { s }_{ n' }-{ s }_{ n } \right) =s\in S and hence 0<s<ε0<s<\varepsilon . This shows that for any ε>0\varepsilon >0, sS\exists s\in S such that 0<s<ε0<s<\varepsilon .

Now let ε>0\varepsilon >0 be a real to be chosen such that ba>ε,a,bRb-a>\varepsilon ,a,b\in R be given. Then it is trivial to show that n1Z\exists {n}_{1}\in Z such that a<n1ε<ba<{n}_{1}\varepsilon <b. Now we take ε=ba2\varepsilon =\frac { b-a }{ 2 } . Then sS\exists s\in S such that 0<s<ε0<s<\varepsilon . Hence, n2Z\exists {n}_{2}\in Z such that a<n2s<ba<{n}_{2}s<b. Since by property ii), n2sS{n}_{2}s\in S,the property is proved.

There is another way to prove the above proposition. First prove that every proper subgroup of the additive group RR is either dense or cyclic in RR. Then prove that the given subset SS of RR is a subgroup of RR that is not cyclic in RR..hence it is dense in RR.

Note by Kuldeep Guha Mazumder
3 years, 10 months ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

There are no comments in this discussion.

×

Problem Loading...

Note Loading...

Set Loading...