# Proving equivalences

• March 29th 2011, 06:03 PM
Connected
Proving equivalences
Let $X\ne\varnothing$ and let $c\in\mathbb R$ an upper bound for $X.$ Prove that the following sentences are equivalent:

a) $c=\sup X.$

b) For all $n>0$ exists an element $x\in X$ so that $c-\dfrac1n

First a) implies b): since $c=\sup X,$ then the number $c-\dfrac1n$ is not an upper bound of $X$ then there exists $x\in X$ so that $c-\dfrac1n Is that enough?

I'm having problems to prove b) implies a), how to proceed?
• March 29th 2011, 06:21 PM
Beaky
Suppose $c \neq \sup X$. Then there exists an upper bound c' < c for X...