  In this paper, we rigorously prove the existence of self-similar converging
shock wave solutions for the non-isentropic Euler equations for $\gamma\in
(1,3]$. These solutions are analytic away from the shock interface before
collapse, and the shock wave reaches the origin at the time of collapse. The
region behind the shock undergoes a sonic degeneracy, which causes numerous
difficulties for smoothness of the flow and the analytic construction of the
solution. The proof is based on continuity arguments, nonlinear invariances,
and barrier functions.
