 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 defineThese are nonempty finite sets, and the transition maps preserve them. The nonemptiness theorem for inverse limits of finite sets supplies a compatible tuple $x\in {\underleftarrow{\lim }}_j{X}_j=G$, for which $x^m=g$. This is the finite-quotient criterion for roots in a profinite group.