What's the idea behind proving $\sup(0,1)=1$?
I made a plan on how to prove such statements for any $\sup M = a$:
(a) $a$ is an upper bound of $M$
(b) $\forall \varepsilon$ $\exists x\in M: x>a-\varepsilon$
a) is easy to prove. $1$ is an upper bound of $x\in (0,1) \Leftrightarrow 0<x<1$.
b) Let $\varepsilon >0$, then we have to show that there exists an $x\in M$ such that $x>1-\varepsilon$ is true. How can I do that? What's the reasoning behind choosing a valid $x$?