Tuesday, April 26, 2011

a proof of least resistance ..

for the final exam in my proofs class, there were five problems. i told my students what two of the questions were, but not exactly.
  1. reproduce five definitions, which will come from certain sections of the textbook (and yes, this is the same kind of problem as before).
  2. 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.
as it happens, the proof of Cantοr's theorem that i gave in lecture was different than the one written in their textbooks, and i told the class that either one is valid for the exam .. and if you want, you can write your own proof.

[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:
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?
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:
  1. 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.)

  2. 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 ..
it's a "geometric" proof, sure, but not the most efficient one. then again, it was the only way i could remember how to do it ..

.. 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).

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