Dense subset of R

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

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

ii)Homogeneity in $Z$: If $x\in S$ and $k\in Z$, then $kx\in S$

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

Also we note that $\forall m,n,m',n'\in Z$, $m+n\sqrt { 2 } =m'+n'\sqrt { 2 } \\ \Rightarrow m=m'\quad and\quad n=n'$

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

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

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

There is another way to prove the above proposition. First prove that every proper subgroup of the additive group $R$ is either dense or cyclic in $R$. Then prove that the given subset $S$ of $R$ is a subgroup of $R$ that is not cyclic in $R$..hence it is dense in $R$. Note by Kuldeep Guha Mazumder
3 years, 10 months ago

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

> 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 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$