In which we think about sequences even when they don’t converge.
- We proved Lemma 2.
- Theorem 3 (Bolzano-Weierstrass theorem): Let be a bounded sequence of real numbers, say for all . Then there is a convergent subsequence: that is, there are and a real number such that as . We saw one proof, using ‘zooming in’ (also known as ‘interval bisection’ and ‘lion hunting’). There was a small typo very near the end: the notes should say “ as ” rather than as .
Understanding today’s lecture
Here are some exercises to get you thinking about what we discussed today.
- Show that every bounded real sequence has a monotone subsequence. Deduce another proof of the Bolzano-Weierstrass theorem.
- When you’ve tried the above exercise, you could try this ‘proofsorter’ activity to consolidate your understanding of the two proofs you’ve seen.
- You could try running each of the two proofs on some examples of sequences, to get a feel for what they tell us. Do they give us a way to find a convergent subsequence, or just tell us that there must be one? If they find convergent subsequences, do they find the same ones?
- Is there an analogue of the Bolzano-Weierstrass theorem for sequences of complex numbers?
I said in lectures that the Bolzano-Weierstrass theorem is extremely useful, but you might be wondering what it’s useful for. We’ll see some applications in lectures later in the course, but in the meantime you might find this article on the Tricki interesting. There are also several links in the summary of the lecture above that you could read.
Preparation for Lecture 3
If a sequence converges, then eventually all the terms get really close to the limit, so they’re eventually all really close together. How could we express that idea more formally? And is the converse true? That is, if eventually all the terms get really close together, does that mean that the sequence must converge?