On the analysis of Christofides's algorithm Suppose you have even number of vertices, they form as a complete graph denote the graph by $G$. Now, suppose we compute the minimum weight perfect matching denoted by $P$ (so $P$ has alternating edges, i.e. one is a perfect matching and one is not along all the path $P$ through all vertices of $G$). Now, suppose $T^*$ is TSP tour of $G$.

Show that the inequality is true:
$$ \operatorname{cost}(P) \le \min \{ \operatorname{cost}(N_1), \operatorname{cost}(N_2) \},$$
where $N_1$ and $N_2$ are any two perfect matchings on $T^*$.

Note that this inequality from the analysis of Christofides's algorithm, see this nice paper.
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 $N$.