Vtyjkyuk

Currently, I am trying to show that the profinite completion $hatmathbbZ$ of $mathbbZ$ is isomorphic to $prod_p mathbbZ_p$ (as topological groups) where $p$ runs through all prime numbers and $mathbbZ_p$ denotes the ring of $p$-adic integers.

By definition, we know that $hatmathbbZ$ is the inverse limit of $mathbbZ/nmathbbZ$ and $mathbbZ_p$ is the inverse limit of $mathbbZ/p^nmathbbZ$. Furthermore, the Chinese Remainder Theorem tells us that if $n = prod_p p^e_p(n)$ denotes the prime factorization of $n$, then we have $$ mathbbZ/nmathbbZ simeq prod_p mathbbZ/p^e_p(n) mathbbZ.$$

So I would like to show that
$$
hatmathbbZ = varprojlim_n mathbbZ/nmathbbZ simeq varprojlim_n prod_p mathbbZ/p^e_p(n) mathbbZ = prod_p varprojlim_n mathbbZ/p^e_p(n) mathbbZ = prod_p varprojlim_n mathbbZ/p^n mathbbZ = prod_p mathbbZ_p,
$$

but I am not sure about these steps, especially with the third and fourth step. Could you please give me an easy explanation whether this works or not? Thank you!