TA: Marcel Goh
Email address: marcel[dot]goh[snail]mail[dot]mcgill[dot]ca
Tutorial time: Thursdays 16:30 to 18:00 in Bronfman 360.
Office hours: Wednesdays 11:00 to 12:00 13:30 to 14:30 15:00 to 16:00
and Thursdays 13:00 to 14:00 in McConnell 311. If neither
of these times works for you and you need to see me urgently, you can email me and make an appointment.
Here are the questions we covered each tutorial, along with their solutions. I will not post versions without the solutions. Just cover them, or make the window really small to show only one question at a time. It is a good exercise in impulse control to try not to peek. That said, it is also a good exercise to try each question for a good five to ten minutes, then uncover one sentence of the solution if you're stuck and try to finish the proof from there, repeating as needed. Once you finally get to the solution by this method, wait an hour or two (maybe go for a walk, get some food, work on your COMP assignment, etc.) so that you've "forgotten" the proof, then try again to reconstruct it, not from memory, but from the understanding you gained by "cheating" the first time.
11 September
We discussed the contrapositive in general, then did a proof by contrapositive. After that, we saw
two proofs by induction and finished off with an interesting proof by cases.
18 September
I was sick so we had a cybertutorial today.
We did a proof involving images and preimages of a function, then dwelt on the definitions of supremum
and infimum for quite a long time. We finished by proving some properties of the absolute value
function.
25 September
Today we did lots of problems involving the supremum and infimum. As dessert, for the last fifteen minutes
we talked about the Cauchy–Schwarz inequality.
2 October
We were only able to do two problems today, one involving the Ruzsa triangle inequality, and the other
proving that the Archimedean property and the nested interval property together imply the least upper
bound property. I prepared notes about Cantor's diagonal argument, which we didn't get to, but as promised
they are in the PDF linked above.
9 October
Today we did sequence problems from the problem sheet, which were boring but instructive. Then I finished
off with trying to prove that if a sequence converges to a limit, then its sequence of averages converges
to the same limit. The proof I'd prepared was wrong, so we figured out
a proper one together on the fly. It's probably
best for you to go over the complete proof in the PDF above though, since I was a bit messy with my
presentation during the actual tutorial.
23 October
Today we did three limit problems. The first was easy. The second was a bit harder: we proved the
quotient law for limits by first proving the reciprocal law. Then we skipped a problem (that I still
left on the sheet, in case you want to do it at home) in order to fit in a proof of Fekete's lemma
(the last problem on the sheet). We talked a lot about how to come up with choices for ϵ and N,
etc.
30 October
Somehow I fell sick again so we did today's tutorial over Zoom. Before starting the tutorial,
we discussed a question from the midterm that not a lot of people got. Then we showed that if
you start with a number bigger than 1 in your calculator and hit the square root button over and
over again, you'll get closer and closer to 1 (but never reach it). Then we proved that the
monotone convergence theorem implies the nested interval property and the Archimedean property,
which, combined with what we showed on 2 October, shows that the monotone convergence theorem
also implies the least upper bound property of the real numbers. We finished off by calculating
some limits related to Euler's number e.
6 November
Today we proved that the Bolzano–Weierstrass theorem implies the monotone convergence theorem,
and then showed that Cauchy's criterion, along with the Archimedean property, can be used to prove
the Bolzano–Weierstrass theorem. This adds two more statements to our list of
Things That Are Equivalent to Completeness. Then we defined divergence to positive and negative infinity,
and proved that the two-point compactification of the reals is sequentially compact.
13 November
We did three routine exercises concerning the rudiments of topology. Then we defined the Cantor
set and discussed it at length. We proved that it is closed, uncountable, and does not contain
any nontrivial intervals.
20 November
Today we warmed up with some routine properties of limits (of functions), and then discussed
two exotic constructions: the Dirichlet function and the stars of Babylon function.
Although you haven't covered continuity yet in the class, what I really proved about the
stars of Babylon function is that it is not continuous at any rational points and yet continuous
at every irrational point on the real line. Interestingly, it is impossible to construct a function
that is continuous at all rational points and discontinuous at all irrational points! (But
the proof of that would take us too far afield from this course.)
27 November
Today we kicked off by proving some basic laws about continuous functions, then used them to show
that there are always two opposing points on the globe that have the same ambient temperature.
Then we showed that the location of roots theorem (a special case of the intermediate value theorem)
implies the least upper bound property. This brings the number of Things That Are Equivalent to
Completeness (that we know of) up to 6 (if I counted right). Then we proved that the "modern"
definition of continuous is equivalent to the Analysis 1 definition of continuous, over the reals.
Lastly we showed that any level set of a function is closed.