Approximation algorithm to visit all nodes in an undirected, weighted, complete graph, with shortest sum of edge weights
Suppose $G$ is a weighted graph, and $T_{OPT}$ is the optimal route you seek. 

For clarity: $T_{OPT}$ is a path that connects all vertices, such that for any other path $P$ that connects all vertices, it holds that $w(T_{OPT}) \leq w(P)$.

Denote $M$ for the MST of $G$. Since $T_{OPT}$ is a subgraph of $G$ that connects all its vertices, we can say:


$w(M) \leq w(T_{OPT})$ - by definition of MST.


Denote $M^*$ An euler tour of $M$. Since $M^*$ visits each edge of $M$ exactly twice, it holds that:


$w(M^*) = 2w(M)$
$M^*$ is some path that visits all vertices of $G$, hence a path that solves your problem (although we cannot say $w(M^*) = w(T_{OPT})$)


Following from 1. 2. and 3. we get:

$$w(M) \leq w(T_{OPT})$$
$$2w(M) \leq 2w(T_{OPT})$$
And, since $w(M^*) = 2w(M)$
$$w(M^*) \leq 2w(T_{OPT})$$