Supremum of a set of irrational numbers I need help with the following example:
Let S be the set of all irrationals in $[0,1]$. Show that $\sup(S) = 1$.
Is there some property that I should be referring to when proving problems like these? The set definition states that it is bounded between $[0,1]$ so is it possible to just say $\sup(S) = 1$ based off that knowledge or is that an insufficient proof?
Thanks in advance.

Observe that $\sup (S) \leq 1$ since $1$ is an upper bound for $S$. If $\sup (S) < 1$. Let $a = \sup (S)$, then $a < 1$. Choose an irrational number $q$ such that $a < q < 1$, then $a$ is no longer an upper bound of $S$. Thus $a = 1$.
