Convex sequence

A sequence of real numbers , satisfying the condition

(*)

Putting

condition (*) may be written as

The geometrical meaning of condition (*) is that the broken line in the ()-plane with corners at the points , is convex. If the sequence is both convex and bounded, then:

1) it does not increase and thus converges to a finite limit;

2) ;

3) .

If is a convex function (of a real variable) for , the sequence , is convex.