ABSTRACT
Strong convergence expresses that the sequence enters the external number in finite time, meaning that from a certain natural number onwards the terms of the sequence are included in the limit set. Due to the fact that the convergence properties may happen in finite time, a refinement is considered, taking into consideration the segment on which the convergence actually happens. This chapter shows that the usual properties of the convergence of sequences remain valid or may be adapted to the context of flexible sequences. It also shows that the usual rules of operations of limits of sequences are valid in the context of definable flexible sequences and for strong convergence. The chapter explains Cauchy’s characterization of convergence for sequences, i.e. the fact that a sequence is convergent if and only if it is a Cauchy sequence, holds for the convergence of flexible sequences.
