# Open Intervals

1. Mar 15, 2013

### JCienfuegos

Since we all agree that 1 = 0.999..., what does PF think the largest number in the open interval (0,1) is?

2. Mar 15, 2013

### micromass

What do you think the smallest number in the interval is?

3. Mar 15, 2013

### pwsnafu

Just because you can ask a question, doesn't mean there is an answer. (0,1) doesn't have a largest element. It's as simple as that.

4. Mar 15, 2013

### JCienfuegos

I always imagined that the smallest number less than one was the infinitesimal 0.9...., but that cannot be, because that equals one. The only other answer I could come come up with was that only terminating decimals of the form 0.999...9 would be in the interval, but that's no good either.

I guess the smallest number would be of the form 0.000.....01 and it too would be non-terminating.

5. Mar 15, 2013

### jbunniii

But that number appears to terminate. Is there anything after the 1?

You are overthinking this. We agree that (0,1) does not contain 0. So suppose it has a minimum, call it M. This is a positive number. Then M/2 is a smaller positive number. This is a contradiction, so there is no minimum after all.

6. Mar 16, 2013

### HallsofIvy

The open interval (0, 1) has NO largest or smallest member. You cannot talk about "the form 0.000.....01 and it too would be non-terminating". If is non-terminating, it does not have a last digit so you cannot assert that it is 1.