  In this paper we investigate convergence for the Variational Iteration Method
(VIM) which was introduced and described in \cite{He0},\cite{He1}, \cite{He2},
and \cite{He3}. We prove the convergence of the iteration scheme for a linear
Klein-Gorden equation with a variable coefficient whose unique solution is
known. The iteration scheme depends on a {\em Lagrange multiplier},
$\lambda(r,s)$, which is represented as a power series. We show that the VIM
iteration scheme converges uniformly on compact intervals to the unique
solution. We also prove convergence when $\lambda(r,s)$ is replaced by any of
its partial sums. The first proof follows a familiar pattern, but the second
requires a new approach. The second approach also provides some detail
regarding the structure of the iterates.
