I've noticed my method for proofs involving limit points isn't being used in the proofs I've read here. I'm curious if there's something I'm missing that makes them incorrect. Here's my proof for $\overline{A\cap B}\subset \bar{A}\cap\bar{B}$.

The problem reduces to showing $(A\cap B)'\subset A'\cap B'$, so let's show this:

Let $x\in(A\cap B)'\Rightarrow \forall B\_r(x), B\_r(x)\cap (A\cap B)\backslash\{x\}\neq \emptyset$

$\Rightarrow (B\_r(x)\cap A)\cap (B\_r(x)\cap B)\backslash\{x\}\neq\emptyset$

$\Rightarrow x\in A'$ and $x\in B' \Rightarrow x\in A'\cap B'$.

I know equality doesn't hold in general, but we can show the opposite since $\cap$ distributes:

Let $x\in A'\cap B'$. Then $\forall B\_r(x), B\_r(x)\cap A\backslash\{x\}\neq\emptyset$ and $\forall B\_s(x), B\_s(x)\cap B\backslash\{x\}\neq\emptyset$. 

Just take $s=r$, then we have $(B\_r(x)\cap A)\cap (B\_r(x)\cap B)\backslash\{x\}\neq\emptyset$.

$\Rightarrow (B\_r(x)\cap A)\cap (B\_r(x)\cap B)\backslash\{x\}\neq\emptyset$

$\Rightarrow B\_r(x)\cap(A\cap B)\backslash\{x\}\neq\emptyset$

$\Rightarrow x\in (A\cap B)'$.

Where am I going wrong?