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.