proof by negation - there is no supremum A = { x + 1/x : x > 0 } How do you prove that there is no supremum for this set? I think this is the inequality needed: M - ε > x + 1/x M - the supremum How do you keep from this step? thanks

Suppose the set $$A=\\{x+1/x:x>0\\}$$ has a supremum. Call it $M$. Then, $$x+\frac{1}{x}\leq M,\quad\forall x>0$$ In particular, take $x=1/n$ for $n\in\mathbb{N}$. Then, for all $n\in\mathbb{N}$, $$\frac{1}{n}+\frac{1}{1/n}=\frac{1}{n}+n\leq M$$ $$\Rightarrow n\leq \frac{1}{n}+n\leq M,\quad\forall n\in\mathbb{N}$$ That is, $\mathbb{N}$ is bounded above by $M$. Contradiction.