A war between proof and intuition - Week 5

Our week 5 material was on proofs and how to rigorously prove
mathematical statements. One example we have seen in class was the direct proof.
Which I personally though was the easiest form of proof.

For example let's try to prove the fact that:
$\forall n\in\mathbb{N}, n \text{ is odd} \Rightarrow n^2$ is odd.

Proof:
Assume $n\in\mathbb{N}$ # $n$ is a typical natural number
    Assume $\exists i \in \mathbb{N}, n = 2i + 1$ # definition of odd.
        Then $n^2 = (2i + 1)^2$  # square both sides
                        $= 2i^2 + 4i + 1$ # Simplify
                 $n^2 = 2(i^2 + 2i) + 1$
        Then $\exists j \in\mathbb{N}, n^2 = 2j + 1$ # $j = i^2 = 2i \in \mathbb{N}$
        Then $n^2$ is odd
        Then $n$ is odd $\Rightarrow$ $n^2$ is ood # implication intro.
Then $\forall n\in\mathbb{N}, n \text{ is odd} \Rightarrow n^2$ is odd.

These are the simplest of proofs. I found the other variations of proofs to be Much harder.
For example I personally struggled with more abstract proofs, like PROOF BY CASES.
Until I finally had to ask for assistance during my professor's office hours.

I also found that preparation for this week's midterm, which went well, also helped in grasping
basic proof structures.

Hopefully practice will resolve the difficulty posed by advanced proofs.
{I'm looking at you proof of infinite primes!}