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**Ragnarok** **Prove that the sum of two integers is also an integer. **

This is from an advanced calculus book. These are the facts presented earlier in the book which I assume are supposed to be sufficient to prove this:

$\displaystyle \mathbb{Z}=\text{the natural numbers, their negations, and zero.}$

$\displaystyle \text{If }m,n\in\mathbb{N}\text{, then }m+n\in\mathbb{N}\text{ and }mn\in\mathbb{N}$.

$\displaystyle \text{If }n\in\mathbb{N}\text{ and }n>1\text{, then }n-1\in\mathbb{N}$.

$\displaystyle \text{If }m,n\in\mathbb{N}\text{ and }n>m\text{, then }n-m\in\mathbb{N}$.

And what we want to show is that if $\displaystyle a,b\in\mathbb{Z}\text{, then }a+b\in\mathbb{Z}$.

So I'm not sure how to do it. Do you have to do cases where $\displaystyle a$ and $\displaystyle b$ are positive, negative, or zero?