These are the course notes for MIT's calculus class, 18.014.
Beginning at the bottom of page 4, the professor launches into a long proof that the set of all integers $\mathbb Z$ is closed under addition.
I don't see why I can't demonstrate it by reasoning the following way. All integers in $\mathbb Z$ are either elements of $P$ (the set of positive integers), the negative of $P$'s elements, or $0$. $P$ is defined as the set of elements common to all inductive sets. So $P$ therefore contains $1$ and $x+1$, for any $x$. Any numbers $a,b$ in $P$ is an integer by definition, and is composed of adding $1+1$ multiple times. Therefore, adding or subtracting $a$ and $b$ is really adding or subtracting $1$ multiple times.
Which means that the set of all integers is closed under addition and subtraction, since you can only get integers.
So why is the level of rigor in the lecture notes necessary?