I am new to writing informal proofs. I am fine with formal proofs, but the transition is jarring. I do understand the necessity of informal proofs. My questions are:

1. How do I figure out which steps to include?

2. How should I format the proofs? Most informal proofs are presented in a paragraph form. Consider instead the proof format below. Is this acceptable? Professors, what would you say to a student who wrote a proof like this? I find this easier to write than paragraphs, since no effort is spent on choice of words.

Prove: $\bigcap \{ \mathcal{P}(X) : X \in A \} = \mathcal{P}(\bigcap A)$.

1. First we prove that $\bigcap \{ \mathcal{P}(X) : X \in A \} \subseteq \mathcal{P}(\bigcap A)$.

1. Assume $y \in \bigcap \{ \mathcal{P}(X) : X \in A \}$.
2. Then, for all $X \in A$, $y \in \mathcal{P}( X)$.
3. Then, for all $X \in A$, $y \subseteq X$.
4. Assume $z \in y$.
5. Then, for all $X \in A$, $z \in X$.
6. Then, $z \in \bigcap A$.
7. Then, for all $z \in y$, $z \in \bigcap A$.
8. Then, $y \subseteq \bigcap A$.
9. Then $y \in \mathcal{P}( \bigcap A )$.
2. Now we prove that $\mathcal{P}(\bigcap A) \subseteq \bigcap \{ \mathcal{P}(X) : X \in A \}$.

1. Assume $y \in \mathcal{P}(\bigcap A)$.
2. Then, $y \subseteq \bigcap A$.
3. Assume $z \in y$.
4. Then, $z \in \bigcap A$.
5. Then, for all $X \in A$, $z \in X$.
6. Then, for all $X \in A$ and for all $z \in y$, $z \in X$.
7. Then, for all $X \in A$, $y \subseteq X$.
8. Then, for all $X \in A$, $y \in \mathcal{P}( X )$.
9. Then, $y \in \bigcap \{ \mathcal{P}(X) : X \in A \}$.
3. Thus, $\bigcap \{ \mathcal{P}(X) : X \in A \} = \mathcal{P}(\bigcap A)$.

• That is precisely how I write proofs in my personal notes. I find it easier to follow everything, and to spot errors. I also include a precise list of all the hypotheses. In my case, I would've included something like: ### Hypotheses 1. $A$ is a set. I once showed my notes to a Professor and he didn't like it, though. – étale-cohomology Nov 22 '16 at 3:46

I wouldn't be so structured. Being formal is fine, until it makes communicating a proof hard. So, you want to show that $$\bigcap_{X\in A} P(X) = P(\cap A)$$
The left hand side consists of those sets $z$ that are subsets of every set in $A$. Because $\cap A$ is the intersection of all sets in $A$, and because $z$ is a subset of all of the sets in $A$, $z$ is a subset of $\cap A$. Conversely, assume that $z$ is a subset of $\cap A$. This means that $z$ is contained in every set in $A$. That is, $z$ is a subset of every $X\in A$, which means $z$ is in $\bigcap_{X\in A} P(X)$.