A game for two players: you each pick a real number between in the interval [0,1]. Whoever picks the highest number wins. If you both pick the same, you go again.
Play three times. You can't pick the same number more than once.
Is there a strategy?
This is Mur 
@christianp
I would just pick 1 
@mur2501 For your first move, OK. But you can't pick the same number twice ... what do you pick for the second number?
CC: @christianp
@ColinTheMathmo @christianp
For second 0.999999999......
@mur2501 In standard mathematics that's equal to 1, so it's the same number, and it wouldn't be permitted under the rules.
CC: @christianp
@ColinTheMathmo @christianp
It's very close to 1 but it's not exactly equal to one
@ColinTheMathmo
Well yes I have seen various proofs of that and they are not wrong 
Just that the meaning of infinity in mathematics is still very much dynamic and flexible and I think this property also depends on what we think infinity to be 
@christianp
@mur2501 If they are proofs then the thing they are proving is correct. That means that point 9 recurring equals one.
If you want to move into non-standard analysis then you're on your own.
My time doing graduate work and then research in maths gives me a fairly definite view on this. I've tried to share this in the past, to help people understand why maths is the way it is, and people just keep waving their hands and going "infinity is complicated, and you can ..."
Well ...
CC: @christianp
@mur2501 ... these things have been studied and researched for centuries, and in standard maths the conclusion is that point nine recurring equals one.
If you want to develop your own theory for why it isn't, I can point you at non-standard analysis, hyper-reals, and the surreals, and bid you good luck, and fair winds.
CC: @christianp
@ColinTheMathmo
I now dive into the unknown territory 
This would be fun 
@christianp
@mur2501 @ColinTheMathmo @christianp Note that @ColinTheMathmo was very clear when he said "standard mathematics".
We can have a completely separate discussion about hyperreals or even surreal numbers. That's a really interesting discussion on its own, but as much as I am a fan of hyperreal numbers, that was not the topic at hand.
Also, even when discussing hyperreals, you have to somehow argue that 0.999... is equal to 1-ε. And while I've seen that assertion made, I don't think I've seen a convincing argument why that should be the case.
And again, all of that it outside of standard mathematics, so again not in the scope of the original question.
@mur2501 I've had this discussion too many times, and I'm not going to have it again.
Read this:
https://en.wikipedia.org/wiki/0.999...
If you don't understand something then maybe I'll take time to explain it, but I've wasted a lot of time in the past trying to help people see it.
All of modern mathematics says that zero point nine recurring is another representation of 1, just as 6/8 is another representation of 9/12.
CC: @christianp