Question

Let S be defined as S = {x|x^2 < x}. I'm supposed to show that sup(S) = 1. Now I've shown that 1 is an upper bound, but I'm not sure how to show that it's the least upper bound. It makes intuitive sense, but I'm having trouble finishing it off.

Answer 1

show that for any other upper bound \(u\) we have \(u>1\)

Answer 2

suppose there a smaller one, say \(1-\epsilon\) for some \(\epsilon>0\) then arrive at a contradiction

Answer 3

Ah, I think I see it.

Answer 4

probably easier to work with \(x^2-x<0\) rather than \(x^2<x\)

Answer 5

That's where I went to @satellite73 also

Answer 6

that's more clever @satellite73 nice. yes \(x^2-x=x(x-1)<0\) is easier to work with :-p

Answer 7

Thanks very much to both of you!