Let $$ be the group generated by the matrices $=\begin{bmatrix}2&0\\0&1\end{bmatrix}$, $=\begin{bmatrix}1&0\\1&1\end{bmatrix}$, where the group operation is matrix multiplication.
Prove that $$ contains a subgroup $_$ for each $\in \mathbb{Z}$ such that $$ \cdots \subset H_{-2}\subset H_{-1} \subset H_0 \subset H_1 \subset \cdots$$
So this is all that is given and i'm not sure where to start from this question. I know that $ABA^{-1}$ is just B with the bottom right replaced by $1/2$. From this we can replace A with $A^2$ and so on but i am not sure how i would prove this. Can someone help please?