Suppose there's a (wrong) statement: "Every nonempty subset A of reals that is bounded above has a largest element".
I wrongly proved this statement is correct. Where did I get it wrong? Thanks
Suppose $\sup A$ is not in $A$. Then $\sup A - a > 0$ for every $a$ in $A$. Thus, we can find a positive integer $n$ such that $$ n(\sup A - a) > 1 \implies \sup A > a + 1/n > a\:\:\text{ for any }\:\:a \in A .$$ Thus, $a + 1/n$ is an upper bound less than $\sup A$, contradicting $\sup A$ is the supremum.