This is not an answer to your question, but I think this message is very important to say to you, as you mentioned you are an undergraduate interested in mathematics.
My biggest recommendation is to make math interesting. For example, we use a summation rule,
$$ \sum_{i\in I} (f(i) + g(i)) = \sum_{i\in I} f(i) + \sum_{i\in I} g(i) $$
You can ask, how to formally define a (finite) sum over a set of real numbers, and then how to formally justify such an operation. But is that really interesting? You just proved something you always knew to be true, you just proved it in a more rigorous manner. It still seems fairly boring.
As an alternative, you can use (infinite) summations to prove Dirichlet's theorem on primes in arithmetic sequences. This is actually interesting, surprising, and has a long of interesting mathematics along the way that appears throughout number theory.
I noticed that people make a huge mistake when first studying real numbers. They wonder why $a+b = b+a$, whenever $a,b\in \mathbb{R}$. Then they realize they need to define $\mathbb{R}$, rigorously, afterwards they need to define $+$, and proceed to prove that. All of that could be done, but in the end, it does not accomplish anything you did not know before. All you are doing is holding yourself back from getting to the interesting world of analysis. Most people give-up on analysis before they can even get past the real numbers. I think if people just adopted the intuitive view of the reals, and immediately started with analysis, they would be motivated to keep on learning it.
Therefore, my recommendation for you, as an undergraduate, you want to find something interesting in mathematics. Start working on that topic, and instead of trying to understanding everything all at once, you rather focus on learning material you find surprising. You only have a limited amount of time, and you will become a better mathematician by going-forward than by holding yourself back.
