Archive of Putnam problems and solutions: http://kskedlaya.org/putnam-archive/
MATH 140A: Real Analysis
What is it about?
Part A of the sequence first contends with question: what is a real number? In following this course, it will be revealed that there are actually several equivalent answers to this question, whether it be a construction of the real numbers through dedekind cuts or cauchy completion of the set of rational numbers. Later on, the course will provide you some basic ideas in topology and an introduction to an important idea: a metric. See, as it turns out, there are actually many ways to define distance on a set other than the euclidean metric. In having to deal with any arbritrary metric that could possibly be defined, a study of metric spaces and properties derived from subsets of metric spaces are sessed out in brutal, yet interesting detail. Exotic and curious conterexamples, like the cantor set or the discrete metric, will challenge your preconceived notions of how the theory of metric spaces work!
How does this course fit into the bigger picture of Mathematics?
Mathematical analysis and metric spaces is the cornerstone of analyzing properties of sequences, functions, spaces, transformations and equations. Also, the inherent rigour of this course provides you a glance of what it may take to prove ideas in other fields of mathematics.
Where can I apply this material?
- Successive approximations is intimately tied with numerical analysis
- Point-set Topology
How difficult is the material?
There are several sources of difficulty. For instance, this course may be the first time that mathematical rigour is applied to proofs which are not necessarily trivial. Additionally, not all proofs are obvious through ‘plugging’ and ‘chugging’ assumptions until you reach the conclusions. Many times you will have to take a step back and understand the mechanism of how assumptions can lead to a conclusions. Many other times, you might need to rely on a ‘trick’ or a necessary identity in order to complete a proof.