Does this make mathematical sense? For a given set $A$, 
An element such that $a \in A $ exists. 
If $A$ is a set of all natural numbers, then:
$$ a \in A \in \mathbb{N} \subset \mathbb{Z} \subset \mathbb{R}. $$
Would maths normally be written like this, if it is correct? 
 A: You have written:
$$a \in A \in \mathbb{N} \subset \mathbb{Z} \subset \mathbb{R}$$
and told us to assume $a\in A$ and $A=\mathbb{N}$.  Under that assumption, the inclusion $A \in \mathbb{N}$ is incorrect; the set of all natural numbers is not a natural number (sorry I don't have a reference handy for this elementary fact). The other inclusions are correct.  If you replace $A \in \mathbb{N}$ with $A\subset \mathbb{N}$, then everything becomes correct.
A: A couple of things, if $A$ is the empty set doesn't exists any $a\in A$. If $A$ is the set of all natural numbers than you have $A= \mathbb N$. The inclusions are ok.
A: This question is a bit confusing and no it doesn't make a lot of "sense" overall. Especially given that $A$ being defined as the set of all natural numbers means $A\not\in\mathbb{N}$ but that $A=\mathbb{N}$.
As for your question, would maths normally be written like this... yes, those are all valid mathematical symbols and there is some logic to the way you are formulating them.
A: Just correct a typo as OP stated  "A is a set of all natural numbers":
$$a \in A = \mathbb{N} \subset \mathbb{Z} \subset \mathbb{R}$$
