Thesis (M.A.)--Boston University In this paper, several numerical methods, instructive for the calculation of the approximate solutions of differential equations, are exhibited to be convergent. In the first two methods (Picard's Method and the Cauchy-Euler method), the theoretical importance of numerical solutions is demonstrated by establishing existence and uniqueness theorems for the linear differential equation of the first order dy/dx = f(x,y) subject to the following conditions: The equation is considered in some region of xy space containing a point (xo,yo) and in addition to being continuous, f(x,y) is assumed to satisfy a Lipschitz condition with respect to y, i.e. |f(x,y1)-f(x,y2)| < k|y1 - y2| where k is called the Lipschitz constant TRUNCATED