Solving the Helmholtz equation with iterative methods is challenging because of its indefinite nature and highly oscillatory solutions. The WaveHoltz algorithm mitigates the difficulties of solving the Helmholtz equation by repeatedly solving the wave equation over short periods in time. In this paper, we prove that for stable semi-discretizations of the wave equation, the WaveHoltz iteration converges to an approximate solution of the corresponding frequency-domain problem, provided one exists. We present numerical examples in one and two dimensions using finite difference and discontinuous Galerkin discretizations illustrating these convergence results.