# Prove inequality $(a_1+a_2+…+a_n)(\frac1{a_1}+\frac1{a_2}+…+\frac1{a_n})\geq n^2$ when $a_1,a_2,…,a_n$ are positive numbers. [duplicate]

Let's assume that $a_{1},a_{2},...,a_{n}$ are positive. How to prove this inequality:

$(a_{1}+a_{2}+...+a_{n})(\frac{1}{a_{1}}+\frac{1}{a_{2}}+...+\frac{1}{a_{n}})\geq n^{2}$

My effort: I don't know where to begin.

## merged by Alexander Gruber♦Jan 28 at 6:24

This question was merged with Proof that $\left(\sum^n_{k=1}x_k\right)\left(\sum^n_{k=1}y_k\right)\geq n^2$ because it is an exact duplicate of that question.