= Solution
{c}
Conditional on $\mathcal F_n$, put $x=X_n$. The two possible values have conditional mean
$$
(1-x)\frac x2+x\frac{1+x}2=x.
$$
The process is adapted and bounded, hence integrable. Consequently it is a <martingale>. Since $\sup_n\mathbb E|X_n|\leq1$, the <martingale convergence theorem> gives an almost surely finite limit $X_\infty$, necessarily in $[0,1]$.
Moreover $|X_n-X_\infty|^2\leq1$ and these squared differences tend to zero almost surely. The <dominated convergence theorem> gives $\mathbb E|X_n-X_\infty|^2\to0$. Thus the <Bernoulli polarization martingale> converges \b[almost surely and in $L^2$].
Back to article page
