- reproduce five definitions, which will come from certain sections of the textbook (and yes, this is the same kind of problem as before).
- reconstruct one of two theorems: one of them was the Cantοr intersectiοn theorem. on the day i print out the exams, i'll flip a coin [1] which will decide which one appears.

[

`snickers ensued`]

i thought my proof is intuitive, when i lectured it .. but i hadn't seen it in a while. for some reason, i thought it wouldn't be

*that*different from the book proof ..

.. until i started grading the exams, and realised:

to explain, i was trying to motivate the theorem at the time, why anyone should believe that it's true [2]. so i indicated it in two steps:wtf? this thing is a monster!

why didn't i just give a nice, short proof ..?

.. ye gods, did i prove this on the fly?

i'm pretty sure that i wrote up notes for that lecture.

was i trying to be cute?

- nested intervals imply monotone sequences of the corresponding endpoints. if you take infιmum and suprεmum (respectively, of left- and right-endpoint sequences) then you get a
__possible__interval;

(in other words, do what the picture tells you to do, but do so rigorously.)

- as long as we can prove that this "limiting" interval is nonempty, we're done. this only happens if inf < sup ..

.. but that can't happen, if you treat inf and sup like limits. to do this rigorously, use an ε-closeness argument ..

.. so props to the students who actually proved it that way, on the exam!

on a related note, some students actually gave their own proofs .. distinctly different, too, which was quite cool.

**(-:**

[1]

*as it happens, it landed heads.*

[2]

*i spent an inordinate effort, all this term, trying to make some of the tricker theorems intuitive. as i told the class: "if you don't have an intuitive idea of why the statement is true, then odds are good that you won't be able to prove it." in other words,*

(logic) + (intuition) = (maths).

(logic) + (intuition) = (maths).

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