Is this proof correct and written in a understandable fashion? Given a function $f(x,y) = \sqrt{(x_{1}-y_{1})^2 + (x_{2}-y_{2})^2 +...+(x_{n}-y_{n})^2}$ where $x,y$ $\in$ $\unicode{x211D}^n$, $f:\unicode{x211D}^n \times \unicode{x211D}^n \rightarrow \unicode{x211D}$
Claim: $f(x,y) = 0 \Leftrightarrow x=y$.
I tried proving the forward implication but I'm unsure whether I've communicated my proof in an understandle fashion.
Proof:
"$\Rightarrow$"
Suppose $f(x,y)=0$ $\Rightarrow$ $\sqrt{(x_{1}-y_{1})^2+(x_{2}-y_{2})^2+...+(x_{n}-y_{n})^2} =0$
$\Rightarrow$ $(x_{1}-y_{1})^2+(x_{2}-y_{2})^2+...+(x_{n}-y_{n})^2 = 0$.
Define $\mu_{n}= (x_{n}-y_{n})^2$. $\forall i \in \{1,2,...,n\}$, $\mu_{i} \geq 0$. Since $\mu_{1}+\mu_{2} + ...+\mu_{n}=0$ $\Rightarrow$ $\forall i \in \{1,2,..,n\}$, $\mu_{i}=0$ $\Rightarrow$ $\forall i \in \{1,2,...,n\}$, $x_{i}=y_{i} \Rightarrow x=y$ 
What parts would you change to make sure people understand what you mean?
 A: This reminds me a little bit of when I first took an analysis course and I was convinced that the best way to write a mathematical proof was purely using logical connectives. I started texing my homework at this point so I can even given an example:
Show $(A-B)\cap C \subseteq (A\cap C)-B$
$$d \in (A-B)\cap C \Rightarrow d \in A \wedge d \in C \Rightarrow d \in A \cap C$$
$$d \in (A-B)\cap C \Rightarrow d \notin B \wedge d \in A \cap C \Rightarrow d \in (A\cap C) - B$$
$$\therefore (A-B)\cap C \subseteq (A\cap C)-B$$
This is correct and there is even a certain elegance to it, but it makes for somewhat strained reading. If I had to write such an argument now it would probably read more as:
Let $x \in (A-B) \cap C$ then $x \in (A-B) \subset A$ and $x \in C$ so $x \in A \cap C$. We also have that $x \notin B$ since $x \in A-B$ so $x \in (A \cap C) - B$. 
Which somehow ends up being shorter and simpler to read than my previous construction. You don't have this problem nearly to the extent that I did. But it does crop in some places, your proof is correct but there are too many logical connectives. It's almost always better to write so or then or thereby then to put a "$\Rightarrow$". Sometimes $\Rightarrow$ and $\forall$ have their places, but its best to use words and of course its a difficult line between overly verbose and too terse. These are all things that get easier then more proofs you write and the more feedback you get. I'd suggest rewriting the proof without using any logical connectives and see how it reads. 
