= Solution
{c}
The <floor function> gives the deterministic bound
$$
0\leq X-X_n<2^{-n}
$$
for every outcome, including $X=1$. Thus \b[all three convergence assertions hold]: there is pointwise, and hence <almost sure convergence>, as well as <convergence in probability> and <convergence in L2>. In detail, for $\varepsilon>0$ the <probability> $\mathbb P(|X-X_n|>\varepsilon)$ is zero as soon as $2^{-n}\leq\varepsilon$; also
$$
\mathbb E[|X-X_n|^2]\leq4^{-n}\longrightarrow0.
$$
The <uniform distribution> even gives the exact <second moment> of the error by integrating over the <dyadic partition>:
$$
\boxed{\mathbb E[|X-X_n|^2]=2^n\int_0^{2^{-n}}t^2\,dt=\frac{4^{-n}}3.}
$$
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