Direct proof of inequality between arithmetic and harmonic mean 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 :) 

So we have to prove that $a_i>0$ gives:
$$\frac{a_1+\ldots+a_n}{n}\leq \frac{n}{\frac{1}{a_1}+\ldots+\frac{1}{a_n}}\tag{1}$$
but by Titu's lemma:
$$\left(\frac{b_1^2}{a_1}+\ldots+\frac{b_n^2}{a_n}\right)\left(a_1+\ldots+a_n\right)\geq (b_1+\ldots+b_n)^2 \tag{2}$$
so $(1)$ trivially follows by taking $b_i=1$. $(2)$ can easily be proved by induction on $n$.
