# sequences and series – Lower bound for sum of reciprocals of positive real numbers

I am reading an article where the author seems to use a known relationship between the sum of a finite sequence of real positive numbers $$a_1 +a_2 +… +a_n = m$$ and the sum of their reciprocals. In particular, I suspect that
$$begin{equation} sum_{i=1}^n frac{1}{a_i} geq frac{n^2}{m} end{equation}$$
with equality when $$a_i = frac{m}{n} forall i$$. Are there any references or known theorems where this inequality is proven?

This interesting answer provides a different lower bound. However, I am doing some experimental evaluations where the bound is working perfectly (varying $$n$$ and using $$10^7$$ uniformly distributed random numbers).