We all know that all the subgroups of the additive group $\mathbb Z$ are $n\mathbb Z$ with $n$ varying over $0,1,2,....$I am new in group theory and have done a proof of that,can someone verify if this is correct?
Suppose $H$ is a non trivial subgroup of $\mathbb Z$.then define $H^{+}=\{\alpha\in H|\alpha >0\}$,which is a nonempty subset of naturals,so by WOP,it has a least element $m$,we claim $H=<m>$,clearly $<m>\subset H$. It suffices to show that there are no elements between $(p-1)m$ and $pm$ for all $p$,say $h \in H$ lies between them,then $h-(p-1)m$ which is also in group $H$ but it lies between $0$ and $m$,but $m$ is the least element of $H^+$,so we get a contradiction,is it correct?
It looks fine to me. It is more usual to use the division algorithm ($n=qm+r$). But your approach works fine.