Tips/Tricks for beginners in Analysis and Recommendation

I'm taking my first course in analysis this semester and I find it too hard/ unapproachable although we have not even covered a lot of syllabus as of yet.

So far, we have done elementary set theory, functions, relations, cardinalities and then moved on to sequences, convergence, monotone sequences, cauchy sequences and limsup and liminf.

I was good(read: not too bad) at all the stuff covered up until we started sequences. I don't know how to study for it. I cover all the lectures and try to understand/grasp the proofs done there; still, I feel it doesn't help a lot. Sometimes, the proofs are too difficult to grasp and sometimes, faced with a theorem to proof, I find it hard to start constructing a proof.

Are there any tips/tricks to think a certain way for the aforementioned topics starting from sequences? Also, how does one get better? Is there anything that can be done to improve your analysis-skills?

Thank you.

I suggest you to find an example verifying the statement of the Theorem; many counterexamples in order to understand concretely how hypotheses are fundamental for giving the Theorem and, then, you can study the proof verifying where the hypotheses have been used. I think that this is the better way to study math.