= Solution
{c}
The <tail sigma-algebra> is
$$
\boxed{\mathcal T=\bigcap_{m\geq1}
\sigma(X_m,X_{m+1},\ldots).}
$$
Fix $S\in\mathcal T$. <independence> of the variables and the <pi-systems> criterion imply <independence> of $\sigma(X_1,\ldots,X_n)$ from $\sigma(X_{n+1},X_{n+2},\ldots)$. Hence $S$ is independent of every finite-head <sigma-algebra> $\mathcal H_n=\sigma(X_1,\ldots,X_n)$.
The increasing union $\bigcup_n\mathcal H_n$ is an algebra, hence a <pi-system>, and generates $\mathcal H_\infty=\sigma(X_1,X_2,\ldots)$. Another pi-lambda argument extends the equality $\mathbb P(S\cap A)=\mathbb P(S)\mathbb P(A)$ from that union to all of $\mathcal H_\infty$. Since $S$ itself belongs to $\mathcal H_\infty$, take $A=S$:
$$
\mathbb P(S)=\mathbb P(S)^2.
$$
Therefore
$$
\boxed{\mathbb P(S)\in\{0,1\}.}
$$
This is <Kolmogorov zero-one law>; no moment assumptions on the variables are needed.
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