In the greedy Traveling Salesman algorithm, the algorithm starts from a starting vertex $v_1 = s$, and in $i$th stage, it goes to the closest vertex to $v_i$ that was not visited yet.
I want to find a counter example that the greedy traveling salesman does not provide any constant factor approximation to the TSP with triangle inequality.
Formally, for any constant $c > 0$, find a complete graph $G$, and positive weights on its edges, such that the weights obey the triangle inequality, and the length of the greedy TSP is by a factor of (at least) $c$ longer than the length of the shortest TSP of $G$.
An intuitive example is, say, if we design the weights of the edges so that the last edge of the greedy path(where it cannot make other choices) is arbitrarily large, but that violates the triangle inequality.
Thanks!