Show that functions in O(1) can't grow faster than their composition with themselves
To complete the first approach and show that $\sqrt{n}\notin O(1)$, you just need to show that the statement "There exist constants $c$ and&nbsp;$n_0$ such that, for all $n\geq n_0$, $\sqrt{n}\leq c\cdot1$" is false. That is, you need to show that, for any $c$ and&nbsp;$n_0$, there is an $n\geq n_0$ such that $\sqrt{n}&gt;c$.  For example, you can take $n=n_0+c^2$ (or $\max\,\{n_0,c^2\}$, if you prefer).

I don't really understand your second approach. If $f(n)=O(1)$, it's perfectly possible that $f(n)\in \omega(f(f(n)))$: take any constant function, for example.