I think sequences and series are quite cool, and they tend to pop up all over the place in other fields, including probability, combinatorics, computer science, and communications.

I'm interested: what all does your proofs class cover? Is it something like a foundations class, where you start off with a few definitions and axioms and prove a bunch of stuff, or does it focus on logic and different proof techniques, or a mix?

Perhaps the best thing I can recommend for this whole "why are things true" business is to find someone else in the class to compare and critique proofs with.

According to the syllabus, we'll be going through Logic, Direct Proof and Proof by Contrapositive, Existence and Proof by Contradiction, Mathematical Induction, Prove or Disprove, Equivalence Relations, and Functions. Some Proofs in Calc.

Bless my HS Calc teacher for getting our toes wet early on since some of this sounds familiar. Just a matter of dusting of memories. My tutor has gone through the class so he's my first line of defense in the comparison department. If not, I'd like to start a study group (especially so if some from my Calc class are in the Proofs as well).

- and they tend to pop up all over the place in other fields

I would not be myself without adding "you forgot science". I think that I have seen a series or two in physics. ;)

By the way, one of my favourite uses of sequences in physics is a proof that no matter how many times you will boost your speed (ν) by some constant (Δν) you are still going to be restricted by the speed of light. That's a pretty cool exercise to solve in class.

- Perhaps the best thing I can recommend for this whole "why are things true" business is to find someone else in the class to compare and critique proofs with.

That's a really cool way to learn proofs. After getting some practice it gets better when you can play the "I may or may not have deliberately made an error, can you spot it?" reversal.

The critical point of no return is when you can do this with category theory :P.