A finite-difference scheme is said to be convergent if all of its solutions in response to initial conditions and excitations converge pointwise to the corresponding solutions of the original differential equation as the step size(s) approach zero.
In other words, as the step-size(s) shrink, the FDS solution must improve, ultimately converging to the corresponding solution of the original differential equation at every point of the domain.
In the vibrating string example, the limit is taken as the step sizes (sampling intervals) and approach zero. Since the finite-difference approximations in Eq. (D.1) converge in the limit to the very definitions of the corresponding partial derivatives, we expect the FDS in Eq. (D.3) based on these approximations to be convergent (and it is).
In establishing convergence, it is necessary to provide that any initial conditions and boundary conditions in the finite-difference scheme converge to those of the continuous differential equation, in the limit. See  for a more detailed discussion of this topic.
The Lax-Richtmyer equivalence theorem provides a means of showing convergence of a finite-difference scheme by showing it is both consistent and stable (and that the initial-value problem is well posed) . The following subsections give basic definitions for these terms which applicable to our simplified scenario (linear, shift-invariant, fixed sampling rates).