Prove integer remains $\mathbb{Z}_{n}$ after Division and Floor
It's a simple exercise in inequality. No need to incorporate modular arithmetic. In particular, what you are asked to prove is that given any $0\le r\lt2^w$ and $0\le p<w$, we have: $$ \lfloor \frac{r}{2^p} \rfloor\le\frac{r}{2^p}\lt\frac{2^w}{2^p}=2^{w-p} $$