Real Sequences

  • Upper and Lower Limits of a Bounded Sequence

    The greatest and smallest limit points of a bounded sequence, as given by the preceding tutorial are respectively called the upper (or superior) and lower (or inferior) limits of the sequence. The upper limit of a bounded sequence is denoted by or . Similarly, the lower limit is denoted by or . Evidently, . For […]

  • Limit Points of a Sequence

    A number is said to be a limit point of a sequence if every neighborhood , of is such that, for infinitely many values of, i.e. for any , , for finitely many values of. Evidently, if, for infinitely many values of then is a limit point of the sequence . As in the case […]

  • Boundedness of Sequences

    such that , . Equivalently, is bounded if such that. Evidently, is bounded if and only if is bounded. Upper and lower bounds (supremum and infimum) of , if exist, are called the upper and lower bounds of the sequence . A constant sequence is obviously bounded. In sequences, terms with equal values can occur. […]

  • Introduction to Real Sequences

    George Cantor (1845—1918), the creator of the set theory, made considerable contributions to the development of the theory of real sequences. He found a firm base for most of the fundamental concepts of real analysis in the sequences of rational numbers. Though his layouts are not convenient in the initial stages, they are quite advantageous […]