If you are working through Analysis I as part of a course, please follow your course’s rules as to whether you are allowed to look up solutions to exercises on the internet.
If you are self-studying through the book (this was how I worked through the book), I strongly recommend writing down your proof on paper before reading mine. Not just “I visualized writing out the proof in my head”, but actually taking out a real piece of paper and using a real pen or pencil to write the proof down. This is even more the case if this is your first real (i.e. proof-based) math book, because you very likely don’t know when a proof is correct. Treat each exercise as a meditation. You will remember far better if you think hard before reading the solution, and it will also be much harder to trick yourself into thinking that you understand something when you don’t. Remember the following quote when you are tempted:
The most impressive quality I’ve seen in mathematicians (including students) is the capacity to call themselves “confused” until they actually understand completely.
Most of us, myself included, are tempted to say we “understand” as soon as we possibly can, to avoid being shamed. People who successfully learn mathematics admit they are “confused” until they understand what’s in the textbook. People who successfully create mathematics have such a finely tuned sense of “confusion” that it may not be until they have created new foundations and concepts that they feel they understand.
Even among mathematicians who project more of a CEO-type, confident persona, it seems that the professors say “I don’t understand” more than the students.
It isn’t humility, exactly, it’s a skill. The ability to continue feeling that something is unclear long after everyone else has decided that everything is wrapped up. You don’t have to have a low opinion of your own abilities to have this skill. You just have to have a tolerance for doubt much higher than that of most humans, who like to decide “yes” or “no” as quickly as possible, and simply don’t care that much whether they’re wrong or right.
I made this website because it was frustrating to work through this book without knowing if I had written a good proof. Now that I am better at mathematics, I want to help other people receive feedback on whether their work is correct. I really want to avoid making it easier for people to trick themselves into thinking they “get it” when they actually don’t. So please use this website wisely!
See also: Importance of struggling in learning.
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