In today’s class, we firstly presented some challenge problems. Among those interesting discussions which impressed me the most is the first challenge problem. This question needs us to apply knowledge of supremum and maximum. These 2 definitions are similar but absolutely different. Supremum is the least upper bound of a set but does not necessarily belong to the set. However, maximum should be picked from a set.

Then we also addressed problems about infimum and supremum. Professor’s comment of this presentation gave me a good insight on how should we prove problems clearly. We should first determine what we want to prove and then use our knowledge to go through the proving process. For example, the second part of the challenge problem2, we want to find a fixed “c” but not some different “c”s for different sets of “a” and “b”.

After those presentations, we continued to talk about density. We proved that Q is dense in R by using Archimedean Property and smartly picking the numerator of the rational number r. Furthermore, we also prove that “There exists a real number who is the square root of 2”. The way we find this real number is to prove it to be the supremum of the set t^2 < 2.

One more funny thing is our class was stopped by the fire drill!:)