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 > 0 such that |f(x)-f(y)| < ε whenever |x-y| < δ. Now suppose x_n be a Cauchy sequence in 𝔻, consider x=x_n and y=x_m in def of U.C. function. Given ε > 0 ∃ a δ > 0 such that |f(x_n)-f(x_m)| < ε whenever |x_n-x_m| < δ. But x_n is Cauchy so ∃ k in ℕ such that |x_n - x_m| < δ for all n, m > k. Hence |f(x_n)-f(x_m)| < ε for all n, m > k. So Uniformly Continuous function always preserve Cauchy sequence.
➤ What can we say about a function if it preserves Cauchy sequences?
One thing is clear i.e. such function need not be uniformly continuous(Take f(x) = x^2 on ℝ). The only thing that we've to worry about is continuity. Let f: 𝔻 → ℝ is a function that preserves the Cauchy sequence. Suppose x_n converges to x in 𝔻. Then clearly x_n is Cauchy. Now the sequence x1, x , x2, x, x_3,... is Cauchy (Why ?), then so is f(x1), f(x) , f(x2), x, f(x3),... . So we've got a Cauchy sequence with a subsequence converging to f(x), this implies all of its subsequences converge to the same point f(x). So we can conclude that f(x_n) converges to f(x) for all x_n conversing to x. Hence by the sequential definition of continuity f is continuous.
➤ 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 > 0 such that |f(x)-f(y)| < ε whenever |x-y| < δ. Now suppose x_n be a Cauchy sequence in 𝔻, consider x=x_n and y=x_m in def of U.C. function. Given ε > 0 ∃ a δ > 0 such that |f(x_n)-f(x_m)| < ε whenever |x_n-x_m| < δ. But x_n is Cauchy so ∃ k in ℕ such that |x_n - x_m| < δ for all n, m > k. Hence |f(x_n)-f(x_m)| < ε for all n, m > k. So Uniformly Continuous function always preserve Cauchy sequence.
➤ What can we say about a function if it preserves Cauchy sequences?
One thing is clear i.e. such function need not be uniformly continuous(Take f(x) = x^2 on ℝ). The only thing that we've to worry about is continuity. Let f: 𝔻 → ℝ is a function that preserves the Cauchy sequence. Suppose x_n converges to x in 𝔻. Then clearly x_n is Cauchy. Now the sequence x1, x , x2, x, x_3,... is Cauchy (Why ?), then so is f(x1), f(x) , f(x2), x, f(x3),... . So we've got a Cauchy sequence with a subsequence converging to f(x), this implies all of its subsequences converge to the same point f(x). So we can conclude that f(x_n) converges to f(x) for all x_n conversing to x. Hence by the sequential definition of continuity f is continuous.
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