Bolzano-Weierstrass theorem
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The Bolzano-Weierstrass theorem in real analysis states that every bounded sequence of real numbers contains a convergent subsequence.
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.
A subsequence is a sequence that omits some members, for instance a2, a5, a13, ...
Here is a sketch of the proof:
- Start with a finite interval that contains all the an. 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 an 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.
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.
The Bolzano-Weierstrass theorem is named after mathematicians Bernhard Bolzano and Karl Weierstrass.
External link
- PlanetMath: proof of Bolzano-Weierstrass Theorem (http://planetmath.org/?op=getobj&from=objects&id=2129) (different proof than the one outlined above)de:Satz von Bolzano-Weierstraß