On limits without epsilons (Q1322620)

The following question is considered: Do the usual ``algebraic'' and ``compactness'' properties for the convergence of real sequences determine the class of convergent sequences uniquely? In particular, do they imply that a sequence between two sequences of the same limit has this limit, too? Based on the theory of sequential convergent spaces (vector \({\mathcal L}^*_0\)-groups) the question is answered in the negative. This solves a problem posed recently by \textit{P. E. Peek} [Real Anal. Exch. 17, No. 2, 751-758 (1992; Zbl 0759.26006)], independent solutions are due to J. Jakubík and the reviewer.