Final Blog post, due on December 5

Which topics and theorems do you think are the most important out of those we have studied?

Over the whole semester there has been many that we have studied. The most important I think is the methods of proving things: induction, contrapostive, direct, etc. Equivalence relations also fall under the most important, along with functions. Of the more recent things, the Schroder-Bernstein Theorem is probably the most important.

What do you need to work on understanding better before the exam? Come up with a mathematical question you would like to see answered or a problem you would like to see worked out.

The most recent things are toughest for me (epsilon delta proofs and infinite series in particular). If I had a choice I would like to work out 12.6 or any of the extra problems at the end of the chapter (12.31-12.34). I recognize that some of these were homework problems but I want to make sure that I understand them. We could also just work out similar ones!

What have you learned in this course? How might these things be useful to you in the future?

I've learned a lot about proving things. All of the theorems and topics that I mentioned in the first part are all things that I've learned and have retained. Also I've learned some neat characteristics of numbers, both even and odd, and many, many other proofs and facts!

These sort of things honestly won't help me too much in my career. However, the general mindset of seeking for things that would make a proof not true very much apply. I can use it in engineering analysis of a given situation or problem. Also logic has already helped a ton in the analysis of circuits.

12.4-12.5, due on December 3

Honestly these sections are still like Hebrew to me. I don't have a real concrete understanding of the general format to prove limits; I scooted by just barely in the last homework. Choosing the epsilon/sigma always throws me. I'm sure I'll get it, but as of right now it's my biggest obstacle to mastering these sections. One thing that doesn't make sense to me is the necessity of condition 2 for continuity when we have condition 3. Is there ever a time when condition 3 (on page 289) is satisfied when condition 2 isn't?

I enjoyed the proofs that were showing the addition of two things (whether they be polynomial or just general functions) have the same limit as that of the individual limits added together. The same also goes for the product of two objects. The use of inequality amazes me. It seriously is like the best thing in a proof. You can typically make something smaller (or larger) that makes your proof not only possible but ten times as simple.