A Study on Nθ-Quasi-Cauchy Sequences

Abstract

and Applied Analysis 3 fix z 0 , z k 1 , . . . , z k n k inEwith z 0 = ηk, z k n k = ξk+1, and |z k i −z k i−1 | < 1/k for 1 ≤ i ≤ nk. Now write (ξ1, η1, z 1 1 , . . . , z 1 n 1 −1 , ξ2, η2, z 2 1 , . . . , z 2 n 2 −1 , ξ3, η3, . . . , ξk, ηk, z k 1 , . . . , z k n k−1 , ξk+1, ηk+1, . . .) . (5) Then denoting this sequence by (αn), we obtain that for any positive integer i there exists a positive integer j such that (ξi, ηi) = (αj−1, αj). The sequence constructed is a quasiCauchy sequence, and it is an Nθ-quasi-Cauchy sequence, since any quasi-Cauchy sequence is an Nθ-quasi-Cauchy sequence. This completes the proof of the lemma. Theorem 3. If a function f defined on an interval E is Nθward continuous, then, it is uniformly continuous. Proof. Suppose that f is not uniformly continuous on E. Then, there is an ε0 > 0 such that for any δ > 0 there exist x, y ∈ E with |x − y| < δ but |f(x) − f(y)| ≥ ε0. For every integer n ≥ 1 fix ξn, ηn ∈ E with |ξn − ηn| < 1/n and |f(ξn) − f(ηn)| ≥ ε0. By the lemma, there exists an Nθquasi-Cauchy sequence (αi) such that for any integer i ≥ 1 there exists a j with ξi = αj and ηi = αj+1. This implies that |f(αj+1)−f(αj)| ≥ ε0; hence, (f(αi)) is notNθ-quasi-Cauchy. Thus, f does not preserve Nθ-quasi-Cauchy sequences. This completes the proof of the theorem. Observing that the sequence, constructed in the proof of the preceding theorem, is also a quasi-Cauchy sequence, we obtain that a real function f defined on an interval E is uniformly continuous if (f(αk)) is Nθ-quasi-Cauchy whenever (αk) is a quasi-Cauchy sequence of points in E. Combining this withTheorem 1, we have that a real function f defined on an interval E is uniformly continuous if and only if (f(αk)) isNθ-quasi-Cauchy whenever (αk) is a quasiCauchy sequence of points in E. Corollary 4. If a function defined on an interval is Nθ-ward continuous, then, it is ward continuous. Proof. The proof follows fromTheorem 3 and [7,Theorem 6] so it is omitted. Corollary 5. If a function defined on an interval is Nθ-ward continuous, then, it is slowly oscillating continuous. Proof. The proof follows fromTheorem 3 and [7,Theorem 5] so it is omitted. It is a well-known result that uniform limit of a sequence of continuous functions is continuous.This is also true in case ofNθ-ward continuity; that is, uniform limit of a sequence of Nθ-ward continuous functions isNθ-ward continuous. Theorem 6. If (fn) is a sequence of Nθ-ward continuous functions on a subset E of R and (fn) is uniformly convergent to a function f, then, f isNθ-ward continuous on E. Proof. Let (αk) be any Nθ-quasi-Cauchy sequence of points in E, and let ε be any positive real number. By uniform convergence of (fn), there exists an n1 ∈ N such that |f(α) − fk(α)| < ε/3 for n ≥ n1 and every α ∈ E. Hence,

Cite this paper

@inproceedings{akalli2014ASO, title={A Study on Nθ-Quasi-Cauchy Sequences}, author={H{\"{u}seyin Çakalli and Huseyin Kaplan and Ziemowit Popowicz}, year={2014} }