Quasi-Cauchy sequences, the functions that preserve them, and a weakening of Bourbaki boundedness (Q6621572)

The paper deals with the notion of Bourbaki quasi-Cauchy sequences in a metric space which is defined as: a sequence \((x_n)\) is said to be Bourbaki quasi-Cauchy in \((X,d)\) if for every \(\epsilon > 0\) there exists \(n_o \in \mathbb{N}\) such that for some \(p \in X\), \(x_n \in B_\epsilon^\infty (p)\) for every \(n \geq n_o\). Further, the authors talk about ``Bourbaki quasi-precompact sets'' which are a generalization of totally bounded sets in the Bourbaki sense. Consequently, it is proved that a subset \(A\) of \((X,d)\) is Bourbaki quasi-precompact in \((X,d)\) if and only if every sequence in \(A\) has a Bourbaki quasi-Cauchy subsequence in \(X\). Finally, some investigation has been done regarding ``quasi-Cauchy Lipschitz functions'' which are defined as: a function \(f : (X, d) \rightarrow (Y,\rho)\) is said to be quasi-Cauchy Lipschitz if for any quasi-Cauchy sequence \((x_n)\) in \(X\) there exists \(\lambda > 0\) such that \(\rho(f(x_k), f(x_{k+1})) \leq \lambda d(x_k, x_{k+1})\) for all \(k \in \mathbb{N}\).