I am currently working through p-adic numbers, an introduction by Fernando Q. Gouvêa.
I understand the result that, if $a$ is a non-zero quadratic residue mod $p$, i.e. there is some $q$ such that $q^2 \equiv a \mod p$, then there is $q_k$ such that $q_k^2 \equiv a \mod p^k$ and $q_{k+1} \equiv q_k \mod p^k$ for every integer $k$. In other words, from $q$ we may construct a $p$-adically coherent sequence $(q)_k$ such that $q_k^2 \equiv a \mod p^k$.
It is intuitive to me that we can construct a $p$-adic number that solves $X^2 = a$, however I'm unsure about the formalisation.
The solution to question 22, which works through such an example, suggests to use truncation. Is this suggesting that if $(q_k)^2 \equiv a \mod p^k$ for all $k$, where $(q_k)$ is the $p$-adic number $q$ truncated to $k$ terms, then $q^2 = a$ ? How may I understand this ?
I understand the text is relatively informal, hence I would appreciate help (or resources) to understand how we deal with infinity in this case.