M = set

Statement: This set does not have a supremum in

Prove with contradiction:

Let's say there is an

Then we can assume each of the following 3 options:

(1)

(2)

(3)

, can't be true because we know

is not in

(won't prove that now..)

Here's where I get stuck..

Let's say there is a

so that

So we want that

Then we say

This applies only if

That's what we had done at uni but I don't understand the last 2 paragraphs..

I know if we square a number that is less than 1 that we get a number that is less than the number we squared so I get it why the 'h' in brackets is replaced by 1.. but how do you know for sure that is still less than 2?