w-VI

Here
Here is
Here is a
Here is a post
Here is a post about
Here is a post about week
Here is a post about week six
Here is a post about week six in
Here is a post about week six in the
Here is a post about week six in the course.

Oh, so we did some proofs about the ``floor'' function. I guess it really drives the point home about requiring a complete and rigorous understanding of the ``floor'' operation before we can start to prove anything about it. Further, it demonstrates the validity of showing intermediate/lesser results before whatever it is that we want to prove. Not that we can always just show a bunch of stuff and accidentally stumble on the right answer, but sometimes we can.

I guess the next point is that disproving something is the equivalent of proving its negation. This is useful for disproving something but also comes back to the concept of proof by contradiction, and maybe even proof by contraposition. I guess the idea is that typically, constructing a single example is easier than demonstrating that something is true for all possibilities. We can use the negation operator to exchange existential for universal quantifiers and we should consider the applicability of the contrapositive and the negation whenever we set out to prove things.

Finally, there is the idea of constructing proofs for limits in the ``epsilon and delta'' sense. Where it is easy to imagine the concept whereby a function continuous at some point has an infinite number of points arbitrarily and ``infinitely'' close to itself, it is possibly more challenging to find what scaling factor is the limit on the relative sizes of epsilon and delta. We can imagine for the function prescribing a line, the slope gives the relationship between epsilon and delta. Where we have anything less trivial, we must then consider how its vertical and horizontal spans compare. For the example of a parabola, however we might realize that we can find an arbitrarily large neighbourhood of any point where the function's slope is arbitrarily large. This is why in the class example, everything is simplified when we restrict the size of the delta that we pick for any epsilon. Regardless of epsilon, we can just pick a maximum size for delta, so that if we consider very large values of epsilon, rather than ``chasing'' the upper limit that we can have on delta for that epsilon, we can choose instead some other small neighbourhood of the point of interest which will easily satisfy our requirements for the limit to converge. The ``art'' in this construction is knowing how to pick a limit on delta to get a ``nice'' relationship (scalar multiplication and/or exponentiation) between epsilon and delta. It just makes for a much more satisfying solution and an easier/simpler ``recipe'' (for delta, given epsilon).

That's about all.

Oh, and I totally got perfect on the midterm, which in some way is tacit permission to continue with this ``style'' of SLOG slogging.

Thanks for sloggin' around town, king without a crown, ground-beef brown, all fall down,

J