Basic Set Theory Proof Verification 
Problem:
$$A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$$

My proof:
Let $x \in A \cap (B \cup C)$. Then $x \in A$ and $x \in (B \cup C)$.
$x \in (B \cup C)$ implies $x\in B$ or $x \in C$.
If $x \in B$, then $x \in (A \cap B)$.
If $x \in C$, then $x \in (A \cap C)$.
Therefore $x\in (A \cap B) \cup (A \cap B)$.
EDIT: Continuing Problem
Let $x\in (A \cap B) \cup (A \cap C)$
Then $x\in (A \cap B)$ or $x\in(A \cap C)$
If $x\in (A \cap B)$, then $x\in A$ and $x\in B$
If $x\in (A \cap C)$, then $x\in A$ and $x\in C$
Either way, $x\in A$ and $x\in  B$ or $x\in C$.
Therefore $x\in A \cap (B \cup C)$
Planning on taking an analysis class next semester. Thought it would be best to start learning it now because I've heard that it'll be a difficult class any tips and tricks would be appreciated.
 A: Your proof is good.
As for analysis tips: Here's a quote from my first analysis lecturer.

"All I've ever done in my entire career is add zero, multiply by one, and use the triangle inequality."

What he means is - when you're faced with something that looks a bit tricky, these are the first things you should try. Often there will be some fraction $\frac{a}{a}$ that you can multiply your expression by, and it makes everything a lot simpler to work with. Similarly, adding $(a-a)$ to an expression (e.g. the usual proof of the product rule) can simplify things greatly.
Personally what most people seem to struggle with is the entirely different approach in terms of intuition (compared to algebra). In algebra, to show that two objects are equal, you subtract them and show that the result is $0$. In analysis, to show that two objects are equal, you subtract them and show that the difference is smaller than any arbitrary real number. Many people find this sort of reasoning "fake", and say that they don't really believe it. The real reason they say that is because they don't have an intuition for it. For me the nice part is that I find it all very intuitive, because everything in analysis can be drawn. Whenever you're about to start a proof, try drawing a diagram of what you're proving and it will make a lot more sense.
tl;dr For analysis, add $0$, multiply by $1$, and draw pictures. Your set equality proof is fine.
EDIT
I liked your proof because you used words. People say "a picture is worth a thousand words", but my personal belief is "a word is worth a thousand maths symbols". This becomes especially true later on when you start to use higher level objects (in the sense that they have "big" definitions). A lot of people in my first analysis class struggled with using words because they felt that it wasn't as formal. This is entirely not true. Anywhere in maths words have precise definitions. (If they don't, you just define them before using them.) Then a proof with words ends up a lot more readable and cleaner than any proof trying to just use symbols. You seem to be fine with this already so keep it up!
