Use the hint. That is, suppose we are given an instance $G$ to the Hamiltonian cycle problem. Construct a complete graph $H$ on the $n$ vertices of $G$ and weight these edges so that an edge of $H$ also in $G$ has cost $1$ but an edge of $H$ not in $G$ has cost $1+\delta n$. Note that if we include even just one of these edges in a cycle, then the cost of that cycle is at least $n+\delta n=(1+\delta )n$. Thus, if we ask for a strict $(1+\delta )$-approximation of a TSP of cost $n$, this can only be achieved using original edges of $G$ that form a Hamiltonian cycle.