I want to prove that for any $a,b\in\mathbb{N}$ that $a+b\in\mathbb{N}$, where $0\in\mathbb{N}$. Here is my attempt with induction:
We fix $b$ and induct over $a$. It is true for $a=0,$ since $b+0=b\in\mathbb{N}$. Now suppose for $a\in\mathbb{N}$ that $a+b\in\mathbb{N}$. Since $b$ is arbitrary, we can choose the integer $b+1\in\mathbb{N}$, and we know $a+(b+1)\in\mathbb{N}$ by hypothesis. From this we know $(a+1)+b\in\mathbb{N}$, so the statement is true for $a+1$ as well. Therefore, it is true for all $a\in\mathbb{N}$.
I don't like how I justify choosing $b+1$ to add to $a$. I can't say "choose $b=b+1$", but I'm not sure I can use a different variable name either, since I'm relying on $a+b\in\mathbb{N}\implies a+b+1\in\mathbb{N}$, with the same variable names. How can I write this proof better?