Solution
= Solution
Necessity follows by applying $p_j$ to an equality $x^m=g$. Conversely, suppose every finite quotient contains an $m$th root of $p_j(g)$ and define
$$
X_j=\{x_j\in G_j:x_j^m=p_j(g)\}.
$$
These are nonempty finite sets, and the transition maps preserve them. The <nonemptiness theorem for inverse limits of finite sets> supplies a compatible tuple $x\in\varprojlim_jX_j=G$, for which $x^m=g$. This is the <finite-quotient criterion for roots in a profinite group>.