Bolzano-Weierstrass theorem - Definition

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 |a_n| is less than L for every index n. Graphically, this can be imagined as points a_i 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 a_n. 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 a_n 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.

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.

External link

• PlanetMath: proof of Bolzano-Weierstrass Theorem (http://planetmath.org/?op=getobj&from=objects&id=2129) (different proof than the one outlined above)