Inequality : $\sum_{k=1}^n x_k\cdot \sum_{k=1}^n \frac{1}{x_k} \geq n^2$ I have to show the inequality of $$\left(\sum_{i=1}^n x_i\right)*\left(\sum_{i=1}^n \frac{1}{x_i}\right) \geq n^2.$$For $x_1, ... x_n \in \mathbb{R_{&gt;0}}$ and $ n \geq 1$. I wanted to show this inequality by induction. The basis is clear, but I am not sure how to do the inductive step. Can someone help me with that?
Expand it out as $\sum_{i,j} x_i/x_j$, and use $t + 1/t \ge 2$ for $t &gt; 0$
By C-S
$$\sum_{i=1}^ns_i\sum_{i=1}^n\frac{1}{s_i}\geq\left(\sum_{i=1}^n\sqrt{s_i}\sqrt{\frac{1}{s_i}}\right)^2=n^2.$$

The Cauchy-Schwarz inequality it's the following.

For all reals $a_i$ and $b_i$ the following inequality holds.
$$\sum_{i=1}^na_i^2\sum_{i=1}^nb_n^2\geq\left(\sum_{i=1}^na_ib_i\right)^2.$$
In our case $a_i=\sqrt{s_i}$ and $b_i=\frac{1}{\sqrt{s_i}}.$

Good luck!