I am failing to find a proof for the statement that the p-adic integers have cardinality continuum.
Neal Koblitz's book says that one can find a bijection from $\mathbb{Z}_p$ to $[0,1]$ and then notes that his "bijection" is in fact not injective, which is not really helpful.
The provided function is given by $a_0+a_1p+\cdots\mapsto \frac{a_0}{p}+\frac{a_1}{p^2}+\cdots$. Is there a quick way to use this function to prove that $|\mathbb{Z}_p|=|\mathbb{R}|$?
This function is at least surjective, hence $|\mathbb Z_p|\ge |\mathbb R|$. And the function $a_0+a_1p+\cdots\mapsto \frac{a_0}{(2p)} + \frac{a_1}{(2p)^2} + \cdots$ is injective, hence $|\mathbb Z_p|\ge|\mathbb R|$.