The paper is the English variant, with additional remarks, of two earlier papers published by the author in Romanian, many years ago. It is motivated by the fact that in the last years many mathematicians were interested on the subject and they can’t found copies of acceptable graphic quality.
In the paper we introduce the notions of convex sequence of p order (p being a positive integer) and respectively system of subconvex sequence of first order. One proves that any subconvex sequence of second order and any system of subconvex sequences of first order are convergent.

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Bărbosu, Dan