What is close to $2^{\infty}$ in the 2-adic metric space?
If $2^{\infty}$ isn't defined, where would it seem most appropriate to put it, if it were defined?
I can see that 2-adics are equivalent modulo $2^{\infty}$ which would seem to imply it's at $0$.
$2^\infty = 0$
\begin{align*} \Vert 2^1 - 0\Vert_2 &= \frac{1}{2}\\ \Vert 2^2 - 0\Vert_2 &= \frac{1}{4}\\ \Vert 2^n - 0\Vert_2 &= \frac{1}{2^n}\\ \end{align*}
Thus as $n$ grows $2^n$ approaches $0$, so it makes sense to define $2^\infty = 0$