Proof of Bolzano-Weierstrass Theorem
Instructions
The dynamic GeoGebra applet below can be used to mirror the "construction" process for generating a convergent subsequence of a bounded sequence in the Bolzano-Weierstrass Theorem.
First, you can change the sequence (an). You might try a bounded sequence that converges (given below), or a bounded sequence that diverges (e.g., sin(n) ). Dragging slider "n" will create new terms in the sequence (up to the first 200).
Then, the construction process is as follows (use the GeoGebra activity to mirror this process for the first 5 iterations):
- Choose the Left-half or Right-half of the Interval - but make sure you have chosen one that will have an infinite number of terms in the sequence;
- Pick any term, , in that interval (and such that for all k).
Reflection Questions
Repeating the iteration process of choosing the Left-half or Right-half of the interval that will have an infinite number of terms in the sequence, and picking any term in each interval such that for all . For the intervals , , ,..., what values are in their intersections? Let be a value in all of the intervals, how can we prove the Bolzano-Weierstrass Theorem from there?