Show that $\begin{pmatrix}1 & 1\\ 0&1\end{pmatrix}$ generates an infinity cyclic subgroup in the $\operatorname{SL}(n,\mathbb Z)$ group.
Is it ok to prove that the pattern is $\begin{pmatrix}1 & 1\\ 0&1\end{pmatrix}^k = \begin{pmatrix}1 & k\\ 0&1\end{pmatrix}$ for $k\in \mathbb{Z}$ and then show that these matrices are isomorphic with group $(\mathbb{Z}, +)$?