Summary: Proxy-based sliding mode control (PSMC) is a control scheme proposed a decade ago originally as a position controller for robot manipulators. This controller has a unique mathematical structure that combines a proportional-integral-derivative (PID) controller and a sliding mode controller in an algebraic way, and has demonstrated its practical usefulness in various applications. Its theoretical foundation, however, has been quite immature. This article presents a set of stability proofs on PSMC as a minimum requirement for future practical applications. Finite-time stability and asymptotic stability of terminal attractors are proven with the use of a non-smooth Lyapunov function.