All-order mass renormalization of string theory

String amplitudes for both bosonic and fermionic theories diverges whenever all but one external joining points on the Riemann surface are infinitesimally close and the other is a finite distance away. We show by going to light-cone coordinates that the origin of these divergences is from single-particle intermediate states which have the same free field energy as an external state. An all-order proof is given which demonstrates how to render the amplitudes finite by a mass renormalization. An explicit application at the one-loop level is done