A recent approach uses linear programming (LP) to compute continuous and piecewise affine (CPA) Lyapunov functions for arbitrary switched linear systems. Such a Lyapunov function is a common Lyapunov function (CLF) for all the respective linear subsystems and asserts the exponential stability of the equilibrium at the origin for the switched system. In this letter, we prove that this LP approach is constructive, i.e., that it succeeds in computing a Lyapunov function for the switched system, whenever the origin is exponentially stable.