Bolzano?Weierstrass theorem

The Bolzano?Weierstrass theorem in real analysis states that every bounded sequence of real numbers contains a convergent subsequence.

Related Topics:
Real analysis - Sequence - Real numbers - Convergent

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The sequence a1, a2, a3, ... is called bounded if there exists a number L such that the absolute value |an| is less than L for every index n. Graphically, this can be imagined as points ai plotted on a 2-dimensional graph, with i on the horizontal axis and the value on the vertical. The sequence then travels to the right as it progresses, and it is bounded if we can draw a horizontal strip which encloses all of the points.

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A subsequence is a sequence that omits some members, for instance a2, a5, a13, ...

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Here is a sketch of the proof:

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• Start with a finite interval that contains all the a_n. Since the sequence is bounded, the interval ( -L, L ) which we have from the definition will do.
• Cut it into two halves. At least one half must contain a_n for infinitely many n.
• Then continue with that half and cut it into two halves, etc.
• This process constructs a sequence of intervals whose common element is the limit of a subsequence.

Since every subsequence of a convergent sequence converges, the proof can be generalized to bounded sequences in Rn using induction by considering one component at a time.

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The theorem is closely related to the Heine-Borel theorem. A generalization of both theorems to arbitrary topological spaces is: a space is compact if and only if every net has a convergent subnet.

Related Topics:
Heine-Borel theorem - Topological space - Compact - Net

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The Bolzano?Weierstrass theorem is named after mathematicians Bernhard Bolzano and Karl Weierstrass.

Related Topics:
Bernhard Bolzano - Karl Weierstrass

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