How can I prove that $(a_1+a_2+\dotsb+a_n)(\frac{1}{a_1}+\frac{1}{a_2}+\dotsb+\frac{1}{a_n})\geq n^2$ I've been struggling for several hours, trying to prove this horrible inequality:
$(a_1+a_2+\dotsb+a_n)\left(\frac{1}{a_1}+\frac{1}{a_2}+\dotsb+\frac{1}{a_n}\right)\geq n^2$.
Where each $a_i$'s are positive and $n$ is a natural number.
First I tried the usual "mathematical induction" method, but it made no avail, since I could not show it would be true if $n=k+1$.
Suppose the inequality holds true when $n=k$, i.e.,
$(a_1+a_2+\dotsb+a_k)\left(\frac{1}{a_1}+\frac{1}{a_2}+\dotsb+\frac{1}{a_k}\right)\geq n^2$.
This is true if and only if
$(a_1+a_2+\dotsb+a_k+a_{k+1})\left(\frac{1}{a_1}+\frac{1}{a_2}+\dotsb+\frac{1}{a_k}+\frac{1}{a_{k+1}}\right) -a_{k+1}\left(\frac{1}{a_1}+\dotsb+\frac{1}{a_k}\right)-\frac{1}{a_{k+1}}(a_1+\dotsb+a_k)-\frac{a_{k+1}}{a_{k+1}} \geq n^2$.
And I got stuck here.
The question looks like I have to use AM-GM inequality at some point, but I do not have a clue. Any small hints and clues will be appreciated.

Hint: AM-GM implies
$$
a_1+a_2+\cdots +a_n\ge n\sqrt[n]{a_1a_2\cdots a_n}
$$ and $$
\frac1{a_1}+\frac1{a_2}+\cdots +\frac1{a_n}\ge \frac{n}{\sqrt[n]{a_1a_2\cdots a_n}}.
$$
