  Finding surface mappings with least distortion arises from many applications
in various fields. Extremal Teichm\"uller maps are surface mappings with least
conformality distortion. The existence and uniqueness of the extremal
Teichm\"uller map between Riemann surfaces of finite type are theoretically
guaranteed [1]. Recently, a simple iterative algorithm for computing the
Teichm\"uller maps between connected Riemann surfaces with given boundary
value was proposed in [11]. Numerical results was reported in the paper to show
the effectiveness of the algorithm. The method was successfully applied to
landmark-matching registration. The purpose of this paper is to prove the
iterative algorithm proposed in [11] indeed converges.
