P8 Let U be an open subset of R" and consider a function f : U - R2 with coordi- nate functions f1 : U -> R and f2 : U - R. (i) Let a be a point in the closure of U (i.e., either a E U or a is in the boundary of U). Suppose lima-+a f1() = b1 and lima->a f2(x) = b2. Prove that lima->a f(x) = (b1, b2) directly from the definition of a limit: show that for every E-neighborhood of b = (b1, b2) E R2 there is some 8-neighborhood of a E R" so that f takes every point * * a that is in the intersection of U with Ns(a) to a point in Ne(b) The argument of (i) proves that if the limit of the coordinate functions f1 and f2 exists, then the limit of f = (f1, f2) also exists. (ii) Use (i) to prove that if f1 and f2 are both continuous at a point a E U, then f is continuous at a