Cauchy Sequence and Functions on subsets of ℝ
Hello! Here I am going to discuss some relation between Cauchy Sequences and functions. The main motive is to know what kind of function(continuous/uniform) preserves Cauchy sequences and vice versa. ➤ Does a continuous function always preserve a Cauchy sequence? We know sequential def. of continuity, so if we take a Cauchy sequence x_n on a closed subset of ℝ then it will surely converge to some point x in the domain, hence f(x_n) will converge to f(x). So our main goal is to deal with subsets that are not closed( they must have a Cauchy sequence that doesn't converge). So one such set is (0,1). Now take f(x) = 1/x and consider x_n = 1/n be a Cauchy sequence. f(x_n) = n, which is not Cauchy hence Continuous function need not preserve Cauchy sequence. ➤ Does a uniformly continuous function always preserve a Cauchy sequence? Suppose f: 𝔻 → ℝ is uniformly continuous. By def. of U.C. function, given ε > 0 ∃ a δ ...