I am working on a proof, and I'm stuck with one inequality. I am given that for all $z$ (and large $z$ in particular) and $g$ an entire function, we have $$|g(z)|\leq\sqrt{|z|}+1/\sqrt{|z|}$$
Now I've been working on a contradiction in part of my proof and have arrived at this fact. There exists a $c>0$ such that
$$|g(z)|>c|z|$$
I suspect this is a contradiction when $z$ is large (the $1/\sqrt{|z|}$ term becomes negligible) but I can't seem to find a proof. Any help would be much appreciated. Thank you.