# Intro to Real Analysis: Supremum

## Homework Statement

Find the supremum of E=(0,1)

## The Attempt at a Solution

By definition of open interval, x<1 for all x in E. So 1 is an upper bound. Let M be any upper bound. We must show $$1<=M$$. Can I just say that any upper bound of M must be greater than or equal to one based on the definition of open interval again?

I'm just not sure if that last line is completely correct. Thanks.

## Answers and Replies

s is the least upperbound if for all $\epsilon > 0$, there exists an $a \in (0, 1)$ satisfying $s - \epsilon < a$.