Approximated TSP: weight of minimum spanning tree less than cost of the optimal tour?
If you take any tour and remove any edge, then you get a spanning tree. Hence the weight of a minimum spanning tree is at most the weight of every tour.

I should add that one of the tricks behind this algorithm is that while we can't find a minimum tour efficiently, we can find a minimum spanning tree. Minimum spanning tree is an efficiently computable relaxation of minimum Hamiltonian path.