# Bolzano-Weierstrass theorem

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:

1. 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.
2. Cut it into two halves. At least one half must contain an for infinitely many n.
3. Then continue with that half and cut it into two halves, etc.
4. 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.

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