Theorem 20: the Bolzano-Weierstrass theorem

My first-year students were thinking about the Bolzano-Weierstrass theorem earlier, so it seemed like a natural choice for this week’s theorem.  I’ll try to describe what it says, and then two proofs (since they’re both nice).

The theorem is all about sequences of real numbers.  Here are some examples of sequences:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, …

1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, …

1, -1, 1, -1, 1, -1, 1, -1, 1, -1, …

Let’s think about these sequences for a bit.  In particular, let’s think about whether they have limits, and if so what those limits are. 

Theorem 2: the Intermediate Value Theorem

I recently helped to restore an old mechanical clock to working order.  (There wasn’t much wrong with it, so this was a pretty easy task!)  Having done this, we then needed to regulate the clock: to get it to run at the correct speed.  With a pendulum clock, one does this by adding or removing weights (often pennies) on the bob, which alters the effective length of the pendulum.  This is how the clock known as Big Ben is regulated, for example.  With the sort of clock I worked on, there is a small knob that one can turn (within some limits).

The person with whom I worked on the clock recently sent me an e-mail to report on his experimentation.  At one end of the knob’s rotation, the clock gained something like 40 minutes over the course of a week.  At the other end, it lost about an hour a week.  At that point, the mathematician in me applied the Intermediate Value Theorem to deduce that there must be some position of the knob that would make the clock run at the correct speed.  So what is the Intermediate Value Theorem, and would it also apply to a pendulum clock that is adjusted by adding and removing pennies?