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).