In which we remind ourselves of what it means for a sequence to converge.
- Least upper bound axiom: Let be a subset of that is non-empty and bounded above. Then has a least upper bound.
- Definition of convergence of sequences of complex numbers.
- Definition of increasing, strictly increasing, decreasing, strictly decreasing and monotone sequences of real numbers.
- Theorem 1: Let be an increasing sequence of real numbers that is bounded above. Then converges. You saw this in Numbers & Sets last term.
- Lemma 2: Let and be sequences of complex numbers.
- ‘The limit is unique.’ That is, if and as then .
- ‘Subsequences converge to the same limit.’ That is, if as and , then as .
- ‘Constant sequences converge.’ That is, if for all then as .
- ‘The limit of the sums is the sum of the limits.’ That is, if and as then as .
- ‘The limit of the products is the product of the limits.’ That is, if and as then as .
- ‘The limit of the reciprocals is the reciprocal of the limit.’ If as and for all and , then as .
- ‘Weak inequalities pass to the limit.’ Let be a sequence of real numbers. If for all and as , then .
We’ll prove this next time.
Understanding today’s lecture
To help you to make sure that you have really understood today’s lecture when you work through the lecture notes, here are a few exercises. I strongly recommend trying these (they shouldn’t take too long), as they will help you to keep up with the course. These are the sorts of questions that mathematicians ask themselves when they encounter new ideas. You should train yourself to ask these questions too (if you don’t yet do so), so that eventually you won’t rely on me asking them.
- It would be a good idea to play with some examples to reacquaint yourself with the least upper bound axiom that you first encountered in Numbers & Sets last term. Find some sets that have least upper bounds and some that don’t. How many different reasons can you find for a set not to have a least upper bound? Must the least upper bound necessarily lie in the set/can the least upper bound lie in the set? What about the greatest lower bound?
- Are there any sequences that are both increasing and decreasing?
- Check that you can prove Theorem 1. Can you prove the Least Upper Bound Axiom if you start by assuming Theorem 1?
- Whenever you meet a new result, you should check why the conditions in the statement are there. Are they there because the result would be false without them, or just to make that particular proof work? For example, in Theorem 1 the conditions are that the sequence must be increasing and that it must be bounded above. If we drop either of those conditions, is the result still true? To have a good understanding of a result, one should think carefully about questions like this.
- How much of Lemma 2 can you prove over the weekend? If you are unsure about any of the parts of Lemma 2, try some examples to get a feel for what’s going on. One good strategy is to work through a proof as it applies to a specific example; this can be very helpful for getting an insight into the argument. And again, make sure that you have checked why the various conditions appear.
There are many good introductory real analysis books, and they’re probably all relevant for this course. There are some suggestions in the Schedules, and you could see what your college library has available. Please do share your recommendations (of books or websites) in the comments below.
Preparation for Lecture 2
Pick some examples of convergent sequences and some examples of non-convergent sequences (stick to real numbers). For each one, look at some subsequences. Can you find one that converges? One that does not converge? Several that converge? Several that do not converge? If you can find several that converge, do they necessarily tend to the same limit?
(If the sequence is 1, 2, 3, 4, 5, 6, …, then here are some subsequences:
- 1, 3, 5, 7, 9, 11, …;
- 2, 3, 5, 7, 11, 13, …;
- 1, 2, 3, 10, 10001, 201924792, …;
- 1, 2, 3, 4, 5, 6, ….)
Now try the same questions with some sequences that are bounded (there is some constant so that for all ).