This is homework 1, due Friday September 9 at the beginning of lecture.
We define absolute value as usual: Given a real , we say that is if and is otherwise.
Absolute values have useful properties: for any . Also, iff . The key property is the triangle inequality: .
Formally, a sequence is a function . As usual, we write the sequence as rather than
A sequence is a Cauchy sequence iff for all there is an such that whenever and , we have
A sequence converges iff there is a real such that for all there is an such that whenever and , we have .
Note that these concepts also make sense in . Now we require all the to be rational, and we require and to be rational as well.
- Show that if a sequence converges, then it is Cauchy.
- Give an example of a Cauchy sequence in that does not converge.
- Show that any Cauchy sequence in converges.
Cauchy’s way of defining the reals was to use Cauchy sequences as the basic building blocks rather than cuts. Again, the idea is that we want to have all the limits, and in some of these limits are missing. In the case of cuts, the way of solving the presence of gaps in was by giving names to all the gaps (the cuts), and adding the names. The easiest repair to the lack of limits here will be the same: We give a name to the limits (the sequences themselves) and the reals will be just the sequences.
There is a problem here that does not occur with the construction using cuts, namely different sequences may have the same limit. We should identify all of them.
Recall that an equivalence relation on a set is a binary relation that is:
- Reflexive: For any , .
- Symmetric: For any , if then also .
- Transitive: For any , if and , then .
If is an equivalence relation, the equivalence classes determined by are the sets . An intuitive way of thinking about this is that we are looking at from a distance, and so we cannot distinguish points that are close to one another, we just see them smashed together as a single point. Here, two points are close iff .
Let and be two Cauchy sequences of rationals. Say that iff converges to . Here, of course, is the sequence with .
- Show that is an equivalence relation. Check that any Cauchy sequence is equivalent to infinitely many other sequences.
- Define as the set of equivalence classes of the relation . The elements of are then Cauchy sequences or, more precisely, collections of Cauchy sequences. A typical element of is a class , and we think of as the limit of . Of course, we have a copy of inside : We can identify the rational with the class of all sequences that converge to .
- Define in and verify that with these definitions we have an ordered field.
- Verify that is complete, meaning that the least upper bound property holds.
This gives a second sense in which is complete: It contains the limits of all Cauchy sequences. A small but important point not mentioned above is the following: Given a sequence of rationals, let be its “copy” inside , i.e., . Then is Cauchy iff is Cauchy, and converges to a rational iff converges to .