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?

This is CauchySchwarz 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 righthand side is precisely $\sum x_i$. 

