proofs

alexandrabel90 · #1 $Old$ November 4th, 2009, 09:05 AM

given that S=( 0,1), show that the max S doesnt exist.

to show that, my notes states that:

let M ∈ S and M= max S.

take M* = (M+1)/2

then we need to show that M* ∈ S and that M < M*...

mayb i know if there is a fixed method that i can adopt whener i have to solve proving questions?

Plato · #2 $Old$ November 4th, 2009, 11:13 AM

Quote:

Originally Posted by alexandrabel90 $View Post$

given that S=( 0,1), show that the max S doesnt exist.
to show that, my notes states that:
let M ∈ S and M= max S.
take M* = (M+1)/2
then we need to show that M* ∈ S and that M < M*...

This is always true.
If $a<b$ then $a<\frac{a+b}{2}<b$ .

Bruno J. · #3 $Old$ November 4th, 2009, 11:15 AM

Well suppose there was an $x \in S$ with $y<x$ for all $y \in S$ . Clearly we would need to have $x<1$ , because all elements of $S$ are $<1$ . Thus the open interval $(x,1)$ is not empty, and all of its elements are in $S$ and all of them are greater than $x$ - this clearly contradicts the definition of $x$ .