
Proving equivalences
Let $\displaystyle X\ne\varnothing$ and let $\displaystyle c\in\mathbb R$ an upper bound for $\displaystyle X.$ Prove that the following sentences are equivalent:
a) $\displaystyle c=\sup X.$
b) For all $\displaystyle n>0$ exists an element $\displaystyle x\in X$ so that $\displaystyle c\dfrac1n<x\le c.$
First a) implies b): since $\displaystyle c=\sup X,$ then the number $\displaystyle c\dfrac1n$ is not an upper bound of $\displaystyle X$ then there exists $\displaystyle x\in X$ so that $\displaystyle c\dfrac1n<x\le c.$ Is that enough?
I'm having problems to prove b) implies a), how to proceed?

Suppose $\displaystyle c \neq \sup X$. Then there exists an upper bound c' < c for X...