; such a rectangular path is determined by a succession of consecutive directed vertical and horizontal sides. By induction on , such a path can be deformed to a constant path at a corner of the grid. If the path intersects at a point , then it breaks up into two rectangular paths of length , and thus can be deformed to the constant path at by the induction hypothesis and elementary properties of the fundamental group. The reasoning follows a "northeast argument": in the non self-intersecting path there will be a corner with largest real part (easterly) and then amongst those one with largest imaginary part (northerly). Reversing direction if need be, the path go from to and then to for and then goes leftwards to . Let be the open rectangle with these vertices. The winding number of the path is for points to the right of the vertical segment from to and for points to the right; and hence inside . Since the winding number is off , lies in . If is a point of the path, it must lie in ; if is on but not on the path, by continuity the winding number of the path about is , so must also lie in . Hence . But in this case the path can be deformed by replacing the three sides of the rectangle by the fourth, resulting in two less sides (with self-intersections permitted). Riemann mapping theorem Weierstrass' convergence theorem. The uniform limit on compacta of a sequence of holomorphic functions is holomorphic; similarly for derivatives. This is an immediate consequence of Morera's theorem for the first statement. Cauchy's integral formula gives a formula for the derivatives which can be used to check that the derivatives also converge uniformly on compacta. Hurwitz's theorem. If a sequence of nowhere-vanishing holomorphic functions on an open domain has a uniform limit on compacta, then either the limit is identically zero or the limit is nowhere-vanishing. If a sequence of univalent holomorphic functions on an open domain has a uniform limit on compacta, then either the limit is constant or the limit is univalent. If the limit function is non-zero, then its zeros have to be isolated. Zeros with multiplicities can be counted by the winding number for a holomorphic function . Hence winding numbers are continuous under uniform limits, so that if each function in the