Abstract
The paper deals with generalizations of the classical notion of a limit to (some) divergent sequences of real numbers. The method of arithmetic means provides an example of such an extension of the traditional definition.
More generally, for an infinite matrix A, the so-called A-limitable sequences are introduced, and the Toeplitz-Silverman theorem is recalled as a sample result concerning matrix transformations of sequences. Another type of generalized limit is the Banach limit, which arises from the Hahn-Banach theorem.
Sequences, on which all Banach limits coincide, are called almost convergent sequences. This notion, introduced by G.
G. Lorentz, continues to be a subject of active investigation today.
The relationship between almost convergent sequences and special matrix transformations is also discussed. The exposition is accompanied by comments on the historical development of the subject, basic references to the results discussed, and key sources for the extensive mathematical field of Summability Theory.
Finally, the unusual life story of G. G.
Lorentz is briefly summarized.