Given a finite list $x\_i$ of $N$ positive reals, it seems that $\sum\_{i=1}^N x\_i = \sum\_{i=1}^N x\_i {}^{-1} \Rightarrow \sum\_{i=1}^N x\_i \geq N$. Can anyone give me a proof?



Little conjecture about sums of reciprocals
This is Cauchy-Schwarz inequality. Set $a\_i=x\_i^{1/2}$ and $b\_i:=x\_i^{-1/2}$. Then
$$N=(a,b)\le\|a\|\cdot\|b\|,$$
with equality if and only if $a$ and $b$ are colinear vectors. With your assumption, the right-hand side is precisely $\sum x\_i$.


