A basic question about approximation algorithms for the Traveling Salesman Problem
The misunderstanding comes from two variations of the TSP problem:


The input graph is complete.
The input graph is arbitrary.


The standard proof above must assume the first variation of TSP.  Thus, there is no k-approximation algorithm for the first variation unless $P=NP$.  The same result then follows for the second variation due to the fact that the second variation generalizes the first variation.

To answer the question directly: The input to the TSP variation in question must be a complete graph.

Note: The Wikipedia article states the following: Often, the model is a complete graph (i.e. each pair of vertices is connected by an edge).  If no path exists between two cities, adding an arbitrarily long edge will complete the graph without affecting the optimal tour.