# Showing that Increasing sequences of natural numbers is uncountable

1. Dec 12, 2011

### renjean

1. The problem statement, all variables and given/known data
Show that A, the set of all increasing sequences of natural numbers is uncountable

2. Relevant equations
I know that the natural numbers themselves are countable.

3. The attempt at a solution
I am thinking of using some sort of diagonal argument to prove this.

2. Dec 12, 2011

### mtayab1994

Well take the example:

Let R denote the reals. Let R′ denote the set of real numbers, between 0 and 1, having decimal expansions that only involve 3s and 7s. s. (This set R′ is an example of what is called a Cantor set. ) There is a bijection between R′ and the set S of inﬁnite binary sequences. For instance, the sequence 0101001.. is mapped to .3737337.... Hence R′ is uncountable. Hope this gives you a clue on how to start.

Last edited: Dec 12, 2011
3. Dec 12, 2011

### Dick

You probably know how to show that the set of all infinite sequences of 0's and 1's is uncountable by a diagonal argument. Can you think of a bijection between infinite sequences of increasing natural numbers and infinite sequences of 0's and 1's?

4. Dec 12, 2011

### mtayab1994

That's kind of what I was trying to tell him.

5. Dec 12, 2011

### Dick

Yes, I see that it was. Sorry to be redundant.

6. Dec 12, 2011

### mtayab1994

Not a problem he's here for our help.

7. Dec 12, 2011

### renjean

That helps a lot! Thank you to the both of you.