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  2. Real analysis - Wikipedia

    en.wikipedia.org/wiki/Real_analysis

    Real analysis is an area of analysis that studies concepts such as sequences and their limits, continuity, differentiation, integration and sequences of functions. By definition, real analysis focuses on the real numbers , often including positive and negative infinity to form the extended real line .

  3. List of real analysis topics - Wikipedia

    en.wikipedia.org/wiki/List_of_real_analysis_topics

    Convolution. Cauchy product –is the discrete convolution of two sequences; Farey sequence – the sequence of completely reduced fractions between 0 and 1; Oscillation – is the behaviour of a sequence of real numbers or a real-valued function, which does not converge, but also does not diverge to +∞ or −∞; and is also a quantitative measure for that.

  4. Sequence - Wikipedia

    en.wikipedia.org/wiki/Sequence

    The notion of a Cauchy sequence is important in the study of sequences in metric spaces, and, in particular, in real analysis. One particularly important result in real analysis is Cauchy characterization of convergence for sequences: A sequence of real numbers is convergent (in the reals) if and only if it is Cauchy.

  5. Glossary of real and complex analysis - Wikipedia

    en.wikipedia.org/wiki/Glossary_of_real_and...

    An Introduction to Complex Analysis in Several Variables. Van Nostrand. Rudin, Walter (1976). Principles of Mathematical Analysis. Walter Rudin Student Series in Advanced Mathematics (3rd ed.). McGraw-Hill. ISBN 9780070542358. Rudin, Walter (1986). Real and Complex Analysis (International Series in Pure and Applied Mathematics). McGraw-Hill.

  6. Monotone convergence theorem - Wikipedia

    en.wikipedia.org/wiki/Monotone_convergence_theorem

    In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the good convergence behaviour of monotonic sequences, i.e. sequences that are non-increasing, or non-decreasing.

  7. Limit of a sequence - Wikipedia

    en.wikipedia.org/wiki/Limit_of_a_sequence

    In the real numbers every Cauchy sequence converges to some limit. A Cauchy sequence is a sequence whose terms ultimately become arbitrarily close together, after sufficiently many initial terms have been discarded. The notion of a Cauchy sequence is important in the study of sequences in metric spaces, and, in particular, in real analysis.

  8. Completeness of the real numbers - Wikipedia

    en.wikipedia.org/wiki/Completeness_of_the_real...

    The monotone convergence theorem (described as the fundamental axiom of analysis by Körner [1]) states that every nondecreasing, bounded sequence of real numbers converges. This can be viewed as a special case of the least upper bound property, but it can also be used fairly directly to prove the Cauchy completeness of the real numbers.

  9. Construction of the real numbers - Wikipedia

    en.wikipedia.org/wiki/Construction_of_the_real...

    An axiomatic definition of the real numbers consists of defining them as the elements of a complete ordered field. [2] [3] [4] This means the following: The real numbers form a set, commonly denoted , containing two distinguished elements denoted 0 and 1, and on which are defined two binary operations and one binary relation; the operations are called addition and multiplication of real ...