Does my proof of $|x+y| \le |x| + |y|$ make sense? How do I conclude a proof?

Question

Thank you for reading it. I know I made a lot of mistakes. This is my first ever proof that I have attempted. Another note is that I only have been studying proofs for about a week. Any advice will be helpful.

prove: $|x+y| ≤ |x| + |y|$
Case 1: ∀ values of x<0 and y<0, the function will decrease: $|x+y| \overset{x<0}= |y\pm x|$
$|x+y| \overset{y<0}= |-y+x)|$
$A=|-x+y|$ –-—-> $∂A/∂X=-1$
$B=|-y+x|$ $∂B/∂Y=-1$

Case 2: In the case of (x,y)>0, the two functions opposite of the inequalities are equal.
{|x+y|⇔ |x|+|y|: x>0 and y>0}
This is a normal property of the absolute value theorem.
Notation: {|x+y|∀ values of x and y = |x|+|y| ∀ for all values of x and y}

Case 3: Case 3 proves that the values of |x|+|y| are unaffected by values less than zero
$|x| = \begin{cases} x,&\text{if }x\ge 0\\ -x,&\text{if }x<0 \end{cases}$

$|y| = \begin{cases} y,&\text{if }y\ge 0\\ -y&\text{if }y<0 \end{cases}$

⇔$|X|+|y|>0$ when $(x,y)≠0$

Note: I don’t know if I properly stated the ∀correctly; however, I meant it as “for all“

Thank you for reading it. I know I made a lot of mistakes. This is my first ever proof that I have attempted. Another note is that I only have been studying proofs for about a week. Any advice will be helpful.

***Edited

Answer 1

I think that you are making it much too complicated.

There are only four cases: $x \ge 0$ or $ x < 0$ combined with $y \ge 0$ or $ y < 0$.

Prove the inequality for each of the four cases and you are done.

Concept such as "randomly chosen" values and partial derivatives of the variables are totally extraneous and just get in the way.

As you do more proofs, you will get a feel (from experience) which concepts are relevant and which are not.

Answer 2

The first rule is to make sure the sentences are grammatically correct. Your first sentence starts "As values of x and y are randomly assigned negative values independent of the other variable..." implies that there is another variable beyond just x and y. You then used the implies sign, $\implies$, which doesn't fit in the parenthetical statement. The (x, y)<0 doesn't make mathematical sense. Only real numbers use the sign <, and (x, y) is not a real number. Maybe you meant x times y. Perhaps you could re-structure the first sentence and we could continue afterwards. For a first try you are doing fine.