I am trying to prove the following inequality: for all $n \geq 1$ and all $x_1, \ldots, x_n \in \mathbb{R}^{*}_+$, one has $$ n^2 \leq \left( \sum_{j=1}^n x_j \right) \left( \sum_{j=1}^n \frac{1}{x_j} \right). $$ I tried to use induction: so it is easy to prove the statement for $n=1$. To prove the induction step, let us assume that the statement is true for $n$ and let us prove it for $n+1$.
By noticing that $$ \left( \sum_{j=1}^{n+1} x_j \right) \left( \sum_{j=1}^{n+1} \frac{1}{x_j} \right) = \left( \sum_{j=1}^n x_j \right) \left( \sum_{j=1}^n \frac{1}{x_j} \right) + \frac{1}{x_{n+1}} \left( \sum_{j=1}^n x_j \right) + x_{n+1} \left( \sum_{j=1}^n \frac{1}{x_j} \right) + 1, $$ so it suffices to prove that $$ 2n \leq \frac{1}{x_{n+1}} \left( \sum_{j=1}^n x_j \right) + x_{n+1} \left( \sum_{j=1}^n \frac{1}{x_j} \right), $$ which I am struggling to prove.
