Subset dense in R

  • #1
HamedJafarian
4
0

Homework Statement


How can I prove that the set of rational rational of the form P/2^n for n,p belong to Z is dense in R?


Homework Equations



How can I prove that a set is dense in R?

The Attempt at a Solution


I do not know how to check dense in R!
 
Last edited by a moderator:

Answers and Replies

  • #2
tiny-tim
Science Advisor
Homework Helper
25,838
255
Welcome to PF!

How can I prove that the set of rational rational of the form P/2^n for n,p belong to Z is dense in R?

Hi Hamed! Welcome to PF! :smile:

With questions like this, always start with the definition

what definition has your professor given you for a dense subset?​
 
  • #3
wofsy
726
0

Homework Statement


How can I prove that the set of rational rational of the form P/2^n for n,p belong to Z is dense in R?


Homework Equations



How can I prove that a set is dense in R?

The Attempt at a Solution


I do not know how to check dense in R!

You need to think about how R is defined.
 
  • #4
HamedJafarian
4
0


Hi Hamed! Welcome to PF! :smile:

With questions like this, always start with the definition

what definition has your professor given you for a dense subset?​

Y is a subset of X,Y is dense in X, if for every x that belog to X, there is y blong to Y that is arbitary close to x.
 
  • #5
tiny-tim
Science Advisor
Homework Helper
25,838
255
Y is a subset of X,Y is dense in X, if for every x that belog to X, there is y blong to Y that is arbitary close to x.

ok … then you need to prove that, for any number x in R, there is a number p/2n arbitrarily close to x.

Hint: choose epsilon = 1/2m :wink:
 
  • #6
HamedJafarian
4
0
ok … then you need to prove that, for any number x in R, there is a number p/2n arbitrarily close to x.

Hint: choose epsilon = 1/2m :wink:

Is it correct for when the p ,n are blong to Z?
and with is it m?
 
  • #7
tiny-tim
Science Advisor
Homework Helper
25,838
255
Is it correct for when the p ,n are blong to Z?
and with is it m?

sorry, Hamed, I've no idea what you mean. :redface:

anyway, I'm talking about the standard δ, ε proof … do you know what that is? :smile:
 
  • #8
tiny-tim
Science Advisor
Homework Helper
25,838
255
Hi Hamed! Thanks for the PM. :smile:

(copy my "ε"! :wink:)
HamedJafarian said:
I mean that i must show that for every eps and x, there is a y that y-x<eps.how can i show this one?

Choose m so that 1/2m < ε,

and then … ? :smile:
 

Suggested for: Subset dense in R

Replies
6
Views
355
Replies
61
Views
2K
  • Last Post
Replies
8
Views
853
Replies
8
Views
312
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
1
Views
569
Replies
3
Views
634
Replies
20
Views
729
Replies
12
Views
921
Replies
11
Views
1K
Top