1. ## Question

I didn't know where to ask this question because it's really simple, I uploaded a picture.

It tries to show why the intersection of the all sets from [s,2] were s is defined to be 0<s<1 and it's clear to me that the answer is [1,2]

However when he tries to prove why numbers less than 1 can't be in the intersection he assumes x<1 and then chooses s to be the mid point between 1 and x. and it doesn't satisfy the condition, but I don't get how he can just chose s t be the midpoint of x and 1. if x<1 then s<x<1<2 , you can't just choose s to be the midpoint between 1 and x ...

It's like saying x is less than 2 and then choosing x to be 3 and say it's contradiction. Can someone explain the logic here? Clearly I am not understanding the thing he tried to do.

2. ## Re: Question

He's trying to show that for any x you might choose less than one, there exists an s such that x < s < 1

That should be pretty clear regardless of whatever particular s he might choose by the density of the real numbers. He just happens to pick s as the midpoint.

So there exists an s such that x is not in [s,2] and thus x cannot be in the intersection of all such intervals.