Is there an "elementary" (say ultrafilter-free) proof of the following fact  : if $G$ is a compact (Hausdorff) topological group, if $g \in G$ is any element from this group, and if $P$ is a polynomial with integer coefficients without constant term, then the identity element of $G$ is a limit point of the sequence $n \mapsto g^{P(n)}$.

An other question  : for which integer-valued sequences $u_n$ is the result above still true with $P(n)$ replaced by $u_n$, whatever $G$ and $g$ are  ?

Is there an "elementary" (say ultrafilter-free) proof of the following fact: if $G$ is a compact (Hausdorff) topological group, if $g \in G$ is any element from this group, and if $P$ is a polynomial with integer coefficients without constant term, then the identity element of $G$ is a limit point of the sequence $n \mapsto g^{P(n)}$.

An other question: for which integer-valued sequences $u_n$ is the result above still true with $P(n)$ replaced by $u_n$, whatever $G$ and $g$ are?