= Moving-variable conditional-expectation convergence
Let $(\mathcal F_n)$ be an increasing <filtration> and $\mathcal F_\infty=\sigma(\bigcup_n\mathcal F_n)$. Suppose $X_n\to X$ <almost surely> and $|X_n|\leq Y$ for one <integrable random variable> $Y$. Then
$$
\mathbb E[X_n\mid\mathcal F_n]\longrightarrow\mathbb E[X\mid\mathcal F_\infty]
$$
<almost surely> and in <L1 space>. To prove this, put $Z_n=\sup_{m\geq n}|X_m-X|$. Then $Z_n\downarrow0$, $Z_n\leq2Y$, and $\mathbb E[Z_n\mid\mathcal F_n]$ is a nonnegative <supermartingale> whose <expectation> tends to zero. Its almost-sure limit is therefore zero. This bounds the moving-variable error; the <conditional-expectation convergence along a filtration>, extended from bounded to integrable variables by truncation, handles $X$ itself. <Dominated convergence> also makes the error tend to zero in <L1 space>.
A separate finite bound for each $X_n$ is insufficient. For a <uniform random variable> $U$ on $(0,1)$, take $X_n=n\mathbf1_{\{U<1/n\}}$ and trivial $\mathcal F_n$. Then $X_n\to0$ <almost surely>, but $\mathbb EX_n=1$ for every $n$.
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