Bounded variation and Cauchy sequence

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Difference between Bounded variation and Cauchy sequence

Bounded variation vs. Cauchy sequence

In mathematical analysis, a function of bounded variation, also known as function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. In mathematics, a Cauchy sequence, named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses.

Absolute value

In mathematics, the absolute value or modulus of a real number is the non-negative value of without regard to its sign.

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Banach space

In mathematics, more specifically in functional analysis, a Banach space (pronounced) is a complete normed vector space.

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Bounded function

In mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded.

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Complete metric space

In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M. Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary).

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Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.

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Limit of a sequence

As the positive integer n becomes larger and larger, the value n\cdot \sin\bigg(\frac1\bigg) becomes arbitrarily close to 1.

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Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

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Normed vector space

In mathematics, a normed vector space is a vector space over the real or complex numbers, on which a norm is defined.

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Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

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Sequence

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed.

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Topological vector space

In mathematics, a topological vector space (also called a linear topological space) is one of the basic structures investigated in functional analysis.

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