My idea of this question is to claim $\sup(A)$ and $\inf(A)$ exists (and equals a value) and prove by contradiction that $\min(A)$,$\max(A)$ exists afterwards (and equals $\sup(A)$,$\inf(A)$).
The issue that I have is how to interpret set $A$... I'm not sure what $[0,1] - \{1/n │n \in \mathbb{N}\}$ exactly means so I don't know what to claim $\sup(A)$ and $\inf(A)$ to be and prove that claim.
Some guidance would be greatly appreciated, I am not really sure how to go about proving these four conditions in general.