Theorem of Bolzano-Weierstrass
The theorem of Bolzano-Weierstrass
states that every bounded sequence
of real numbers
contains a convergent
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 which omits some members, for instance a2, a5, a13, ...
Here is a sketch of the proof:
- start with a finite interval which 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 limit of a subsequence.
The theorem is closely related to the theorem of Heine-Borel