The most difficult part of the material for me was Theorem 10.17. Well, I should say that the difficult part was in the proof. The theorem itself appears to be true, upon first reading, and I was surprised to see such an intricate proof for it. I don't understand why the ran f was brought into the proof. Also, I find it confusing when all of this extra notation is involved. Like f^n(x) and B'. It just kind of hurts my head when I read a proof like that. I'd have to reread it many, many times before understanding it, and, even then, would not really get what it is saying. I find it encouraging that we haven't had to prove theorems on exams. That would be really hard.
While the material is interesting and beneficial in the field of mathematics, sometimes I wonder what the application is for me, personally. I appreciate that mathematicians study and enjoy this type of material, but, as I stop to think about my life and career goals, I wonder how much of the material - or let's say, the Schroder-Bernstein Theorem, specifically - will I use later in life. I don't think that I would be cut out to be a math major and study years of mathematics. I have a lot of respect for the professors and other gifted minds in the math department. This material is hard to understand for me. When I feel like the material is going over my head, I think about things like this, to be honest, while I search for motivation to fully understand the material.
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