So let there be the metric space (X, d), let C(N, R) be the set of all convergent sequences in R and let there be the function f: C(N, R) |--> X; f(x_n) --> lim x_n. There's two situations: one where the distance associated with C(N, R) is d_infinity, which is the maximum difference for all elements in the two sequences in C(N, R), and one where the distance associated with C(N, R) is d(x, y) = 1/n where n is the smallest k where x_k != y_k. I need to prove that f is continuous in the first situation and not continuous in the second situation.
So let there be the metric space (X, d), let C(N, R) be the set of all convergent sequences in R and let there be the function f: C(N, R) |--> X; f(x_n) --> lim x_n. There's two situations: one where the distance associated with C(N, R) is d_infinity, which is the maximum difference for all elements in the two sequences in C(N, R), and one where the distance associated with C(N, R) is d(x, y) = 1/n where n is the smallest k where x_k != y_k. I need to prove that f is continuous in the first situation and not continuous in the second situation.