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 $c-\dfrac1n<x\le c.$

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 $c-\dfrac1n<x\le c.$ Is that enough?

I'm having problems to prove b) implies a), how to proceed?