You are Here: Home >< Maths

# Some dense and non-dense sets watch

1. I want to prove the set of reals in [0,1] whose decimal expansion does not contain 3 is not dense.

Here's what I thought: suppose it is dense. Then there is a point of the set in between an open interval where . Then I claim we must have Suppose not: or

The former interval seems to lead to a contradiction without much effort by bounding the positive series above by a negative number. But I can't do it with the latter. Any help?

Got another set, too: prove the set of rational numbers whose denominators is a power of 2 is dense. I have no idea how to begin with this one. Any hints for making it slightly more tractable?

I made an observation (albeit simple and I'm not quite sure if it's true) that we may find a natural number n where 1/2^n < 1/n < ɛ when n > 0 as a simple consequence of the Archimedean property. Perhaps this will be useful? However, I guess I'll have to convert my "every open interval" definition of dense in to terms of ɛ to use it.

Thanks!
2. For the first, note that there is no number in (0.3, 0.399999] whose decimal expansion doesn't include a 3.

For the 2nd, you can easily use your observation to show that for any x in [0,1] we can find a sequence a_n s.t. |a_n / 2^n - x| < 1/n. Then a_n/2^n -> x.
3. You've lost me on the first one I'm afraid, isn't that what I'm trying to do?

### Related university courses

TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: July 23, 2009
Today on TSR

### He lied about his age

Thought he was 19... really he's 14

### University open days

Wed, 25 Jul '18
2. University of Buckingham
Wed, 25 Jul '18
3. Bournemouth University
Wed, 1 Aug '18
Poll
Useful resources

### Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

### How to use LaTex

Writing equations the easy way

### Study habits of A* students

Top tips from students who have already aced their exams