Assignment 2 and proofs in general

The second assignment is turning out to be fairly difficult. The first part is not too difficult and neither is the third part. It's the second part that's pain. I spent a lot of time trying to figure out what the statements even mean. Now I think I might have figured out what they mean, but I'm still not sure whether or not they are true. I think epsilon-delta proofs are the only type of proofs where you can get away without entirely understanding how they work. When I first learned about them, I spent a lot of time trying to understand exactly how the proofs work and why you can set delta to be the minimum of two values. In the end, I never really understood them, but I still managed to get by in tests without truly understanding them entirely. Eventually, doing limit proofs becomes really mechanical. It's just a matter of expressing epsilon in terms of delta and finding a delta that works. You don't really need to understand the thought process behind it. Well, I think that's coming to haunt me now.

So far, I feel this course hasn't really provided me with many tools in doing proofs. All I've really learned is how to write proofs in a more rigorous manner and what you have to show for the different quantifiers. But doing the proof itself isn't any easier. I guess you can't really teach proofs. The part in the middle of the proof, the part that requires a lot of brainpower and creativity, can't really be taught. At most, I think all this course can do is help us expand our thinking when we actually come to that middle part.

On a related note, I think I like logic more. It's fun to work with those symbols and it's easy to understand and it isn't very hard either. My cousin told me that logic courses tend to be fairly easy, and I can see why that's true. This also applies to proving the equivalence of two sentences. All it involves you doing is using a bunch of laws of logic until you can turn one sentence into the other, and hence show equivalence. It's also pretty fun to play around with those symbols.

There are still a few things about proofs that I'm not entirely sure. For instance, you can make delta dependent on epsilon, but why not x? After all, you are saying that for every x, for every epsilon, there is a delta. Hence, shouldn't it also be okay to make delta dependent on x? Also, I'm not entirely sure when I am supposed to indent and leave an indentation block. I'll have to get these straightened out before the test.