Let $f \colon \ [0.1] \to \mathbb R$ is monotonically increasing function and $f(0)>0$ and $f(x)\neq x $ for all $x\in [0,1]$. $$A=\{x\in [0,1] : f(x)>x \}$$
We know: every non-empty subset of $\mathbb R$ bounded above has a least upper bound.
A is non-empty subset of $\mathbb R$ because $0\in A$ and also $A$ is bounded so $\text{sup} A$ exists. Now I want to show that:
- $\text{sup}A\in A$ and consequently $\text{sup}A=\text{max}A$.
- $f(1)>1$
From 1 and 2 conclude that $\text{sup}A=\text{max}A=1$
I am suspicious to own solution. I need the proof in details.
thanks