I need to prove inequality from the title. I know that it follows from $H_n \leq G_n \leq A_n \leq Q_n$, where $H_n, G_n, A_n, Q_n$ are harmonic, geometric, arithmetic and quadratic means of $n$ real numbers, but for some purpose, I need to prove directly that $A_n \geq H_n$.
I have searched and couldn't find anything similar.
EDIT: I forgot to say that I have found this, but it uses Cauchy's Inequality. I would like to find some proof without it, as it exceeds level of the paper I'm writing :)