Is $\overline{\langle2\rangle\cdot(4\Bbb Z+1)}=\langle2\rangle\cdot(4\Bbb Z_2+1)$ in $\Bbb Z_2$?
$\langle2\rangle$ is the set of powers of $2$ and $\cdot$ is the straightforward dot product.
I get that $\overline{(4\Bbb Z+1)}=(4\Bbb Z_2+1)$ where $\Bbb Z_2$ is the 2-adic integers. But what about if we introduce powers of $2$ as factors? Tentatively I would say it seems the given proposition follows because $2b$ is simply the centre of the ball $b$ but then I wonder about the number $0$ which would be in one but not the other, so I'm confused!
Maybe then:
$\overline{\langle2\rangle\cdot(4\Bbb Z+1)}=\overline{\langle2\rangle}\cdot(4\Bbb Z_2+1)$ in $\Bbb Z_2$?