Prove that $\sup[0,1] =1$ Alongside the question in the title, does it matter if the question is $(0,1)$ or $[0,1)$?
I know that it satisfies the first condition, $1$ is an upper bound but I am not sure where to go from there.
Thanks.

Hint: To show that the supremum of $[0,1]$ is $1$, you need to show two things: 1) that it is an upper bound and 2) that there is no lower upper bound.
You say you've already proved (1); so, it comes down to (2).  Can you show that if $x<1$, then $x$ is not an upper bound on $[0,1]$?
