So i have to prove that the relation $\leq$ is a partial order on N, where $≤$ is defined as follows:
$\forall(m_{\in\mathbb N})(n_{\in\mathbb N}) : (n\leq m)\iff (\exists (k_{\in N}) n+k=m)$.
For this one I do understand why it is Antisymmetric and reflexive, but I don't completely understand how it's transitive. For a transitive relation $R$, if $(a,b)\in R$ and $(b,c)\in R$ , then $a$ should also be in a relation with $c$, I just don't know how to prove it on this exact example
Let $l\leq n$ and $n\leq m$.
Then $r,s\in\mathbb N$ exist with $l+r=n$ and $n+s=m$.
From this we conclude directly that: $$l+(r+s)=(l+r)+s=n+s=m$$ and this of course with $r+s\in\mathbb N$ so that the conclusion $l\leq m$ is justified.