= Solution
{c}
The <martingale> is bounded in $L^2$, so the increment-square series from part (a)(ii) is finite. The identity just obtained implies $\sum_n\mathbb E[X_n(1-X_n)]<\infty$, hence $\mathbb E[X_n(1-X_n)]\to0$. By <almost sure convergence> and the <dominated convergence theorem>,
$$
\mathbb E[X_\infty(1-X_\infty)]=0.
$$
The integrand is nonnegative because $X_\infty\in[0,1]$. It must vanish almost surely, giving $X_\infty\in\{0,1\}$.
The <martingale> has constant expectation $a$. Its $L^2$ convergence implies $L^1$ convergence by <Cauchy-Schwarz inequality>, and hence $\mathbb E X_\infty=a$. Therefore
$$
\boxed{\mathbb P(X_\infty=1)=a,\qquad\mathbb P(X_\infty=0)=1-a.}
$$
The limiting law is the <Bernoulli distribution> with parameter $a$.
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