Week 10-Proving General Statements

        From this week and from last week, I learned proofs that involve the definition of lower_bound, upper_bound and limits. Such proofs are very specific about the functions involved in the definitions, and  a function has to be given in order for the proof to work. However, this week, I learned how to prove general statements that does not involve a specific function. In other words, I need to proof certain claims for a general function f and g. What makes this week's material easy is that all these proofs involve similar concepts and formats. The only important thing to keep in mind is to memorize the definition of upper-bound, lower-bound, limits and their corresponding negations. Here is an example of a proof for a general statement:

To begin the proof, I first assume that f, and g both belong to the set of function F. This is because the proof asks me to prove the general statement for all f and g in F. Then I proceed by assuming the antecedent as usual. And from the antecedent, I know that f is lower-bounded by g, hence I can state its definition in the next line. Because I have to assume the antecedent and arrive at the consequent, I have to arrive at the definition that f^2 is lower-bounded by g^2. Hence I have to prove that

is true (where f(n) is f^2 and g(n) represents g^2). From this understanding, I have to pick values for both c and B. Since I don't really know how the proof will turn out, I left the pick c and pick B line empty for now and start by assuming the domain and the antecedent in the following line. Next, I squared f. Since from the antecedent, I know that f is greater than or equal to c'g, I also know that its square is greater than or equal to the square of c'g. In order to use the definition of f lower-bounded by g, n has to be greater than B', hence B can equal to B'. Now that I know what to pick for B, I can go back and write pick B= B'. And because B' is a natural number, B is also a natural number. Then I proceed with my previous calculation and got to that f^2 is greater or equal to c'^2 times g^2. In order to arrive at the consequent that f^2 is greater than or equal to cg^2, all I have to do is make c = c'^2. Hence I know what values to pick for c. And because c' is a positive real number, and real numbers are closed under multiplication, I can state that c is also a positive real number. Now that the proof is finished, all i have to do is to state the bookends and conclude the original claim.

        In conclusion, it is much faster to figure out the values for c and B before tackling the problem. To be able to do this, quickly formulate the proof on a piece of scrap paper. Hence, by figuring out what I have to do in order to arrive at the consequent, I can write up the proof much faster than having to change my c and B values constantly.