Show that there exists a finite disjoint collection $\{E_k\}_{k=1}^M$ of boxes such that $\bigcup_{i}^N R_i = \bigsqcup_{k=1}^M E_k$. 
Let $\{R_i\}_{i=1}^N$ be finite collection of $n-$dimensional boxes formed by the cartesian products of intervals of $\Bbb R$. Show that there exists a finite disjoint collection $\{E_k\}_{k=1}^M$ of boxes such that $\bigcup_{i}^N R_i = \bigsqcup_{k=1}^M E_k$.

I’ve been stuck with this for a good while now. From wikipedia the definition for $\bigsqcup_{k=1}^M E_k $ seems to be that $\bigsqcup_{k=1}^M E_k = \bigcup_{k=1}^M \{(x,k) : x \in E_k \}$, which I don’t really understand.
I tought that I could approach this by elementary set theory just by looking at the elements of either set individually, but I didn’t get anywhere.
If I pick $R_i \in \bigcup_{i}^N R_i $, then $R_i = I_1 \times I_2 \times \cdots$, but I don’t see how I can show that this $R_i \in \bigsqcup_{k=1}^M E_k $?

"All you have to do" is to sort out the elements, which accour in several boxes.
This might be a little tedious, but is not particularly difficult.
Maybe look at a two dimensional picture, and try to figure out how to "disjointinize" your boxes.
If we had a finite collection of sets $A_1,\dotso, A_n$, we could look at the following sets:
$B_1=\emptyset$, $B_2=A_1$, $B_{i+1}=B_i\setminus \cup_{k=1}^{i-1}B_k$.
These sets would be disjoint, but $\bigcup A_i=\bigcup B_i$.
The problem with boxes is, that this construction does not work anymore, because when you intersect a box (which are rectangles), with a box, then the new set does not have to look like a rectangle anymore.
It might look like an L-shape.
But these shapes are now again just made up of boxes.
So the general method of how to do this, should be clear.
What the best way is to perform these constructions, might be not so clear.
Maybe inductively.
