f-rough Cauchy sequences

Suppose x = {x_n } _{n ∈ℕ} is a bounded sequence in some finite dimensional normed space X. Let C denote the cluster point set of this sequence. Then, D_X (C) is the minimal Cauchy degree and r_X (C) is the minimal convergence degree of {x_n } _{n ∈ℕ}, where

That means is a ρ-Cauchy sequence}, and

Suppose ρ ≥ 0 and {x_n } _{n ∈ℕ} is a ρ-Cauchy sequence in some normed space X. Then, {x_n } _{n ∈ℕ} is r-convergent for every r > 2⁻¹ J(X)ρ. If X is finite dimensional then {x_n } _{n ∈ℕ} is r-convergent for every r ≥ 2⁻¹ J(X)ρ.

If x = {x_n } _{n ∈ℕ} is a sequence in a finite dimensional normed space ℝ ⁿ , then