SLOG #5

    This week's material is once again focused on proofs. I found that most of the material was easy to follow and understand, though I may experience difficulties writing them out on my own. However, I still have trouble trying to figure out exactly what's going on with one of the examples in lecture.

    The problem I'm referring to is to prove that for all e in the set of positive real numbers, there is a d in the set of positive real numbers, that for all x in the set of real numbers, if |x - 3| < d, then |x^2 - 3^2| < e. I'm confused as to what the question is asking because there seems to be no relationship between e and d. How is it that the absolute value of (x - 3) is smaller than d implies the absolute value of x^2 - 3^2 is smaller than e? To understand this question better, I tried to look at it like a limit question, where I pick e first as a upper and lower bound, and then I must choose a d that is within the bounds, but not the reference point itself. This perspective helped me a little in perceiving the problem. I can sense a relationship between d and e now; I have to define a d such that for all the possible values of e, whenever |x-3| < d is true, |x^2 - 3^3| < e must be true, which means d has to be some value that depends on e in order to make the implication true. The next part that I am confused about is the content of the proof. Due to my lack of practice with absolute value inequalities, I'm not comfortable in taking numbers out of them. Specifically, I don't understand why d * |x + 3 - 3 + 3| is less than or equal to d* (|x - 3| + 6), because it looks to me that they produce the same value, so they should be equal. Also, the professor arbitrarily chose a number (1) that d is no larger than, so eventually was defined as the minimum between 1 and e/7. Does that mean I can choose any arbitrary number I like, as long as I change the denominator?

    There are still some questions left in my mind, and I will seek the answers during next week.