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? 

Just correct a typo as OP stated  "A is a set of all natural numbers":
$$a \in A = \mathbb{N} \subset \mathbb{Z} \subset \mathbb{R}$$
