In Shilov'sElementary Real and Complex Analysis, he writes on page 216:

'As we know, a (complex) power series a0 + a1 (z - z0) + a2 (z - z0)^2 + ... with radius of convergence p may or may not converge at points on the boundary of its region of convergence, i.e., at points of the circle |z - z0| = p. However, if the series converges at a boundary point z1, then it converges uniformly on the whole segment going from the center of the circle z0 to the boundary point z1. To see this, we need only consider the case z0 = 0, z1 = t1 > 0 (here z1 = t1 is real, as opposed to the general case where it is complex). Why?"

He then goes on to prove the special case with z0 = 0 and z1 = t1 > 0. Why does having proved this special case imply the general case for any point on the boundary of a region of convergence centered at any complex point? I think it may have something to do with shifting the power series and dividing/multiplying, but I'm not sure quite how to make it work.