On the analysis of Christofides's algorithm
Since $P$ is a minimum weight perfect matching, by definition
$$ \operatorname{cost}(P) = \min_N \operatorname{cost}(N), $$
where $N$ goes overl all perfect matchings. In particular, $\operatorname{cost}(P) \leq \operatorname{cost}(N)$ for any particular perfect matching $N$.