# proofs

• Nov 4th 2009, 07:05 AM
alexandrabel90
proofs
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?
• Nov 4th 2009, 09:13 AM
Plato
Quote:

Originally Posted by alexandrabel90
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 $\displaystyle a<b$ then $\displaystyle a<\frac{a+b}{2}<b$.
• Nov 4th 2009, 09:15 AM
Bruno J.
Well suppose there was an $\displaystyle x \in S$ with $\displaystyle y<x$ for all $\displaystyle y \in S$. Clearly we would need to have $\displaystyle x<1$, because all elements of $\displaystyle S$ are $\displaystyle <1$. Thus the open interval $\displaystyle (x,1)$ is not empty, and all of its elements are in $\displaystyle S$ and all of them are greater than $\displaystyle x$ - this clearly contradicts the definition of $\displaystyle x$.