Problem sounds easy enough - prove that if $m$ is in set of all natural numbers (let's call it $\mathbb N$) and so is $n$ than $m+n$ also must be there. Probably it should be done using induction. But how to do it formally? I mean I could tell that $\mathbb N$ is a set $\{1, 1+1, 1+1+1, \dotso\}$ and $m$ is $m$ ones and $n$ is $n$ ones so $m+n$ is $m+n$ ones so it's still in $\mathbf N$ but I doubt that my professor would accept such a proof. It's so intuitive that I don't even know where to start. What should I do?