[proofplan] The proof uses [Doob's Upcrossing Inequality](/theorems/1156) and the [Convergence Criterion via Upcrossings](/theorems/1155). For each pair of rationals $a < b$, the $L^1$-boundedness of the supermartingale provides a uniform bound on the expected number of upcrossings, which by the [Monotone Convergence Theorem](/theorems/509) implies that the total number of upcrossings $N([a, b], X)$ is a.s. finite. A countable intersection over rational pairs, combined with the upcrossing convergence criterion, gives a.s. convergence. Integrability of the [limit](/page/Limit) follows from [Fatou's Lemma](/theorems/510). [/proofplan] [step:Bound the expected total number of upcrossings using $L^1$-boundedness] For rationals $a < b$, let $N_n([a, b], X)$ denote the number of upcrossings of $[a, b]$ by time $n$. The [Doob's Upcrossing Inequality](/theorems/1156) gives: \begin{align*} (b - a) \, \mathbb{E}[N_n([a, b], X)] \leq \mathbb{E}[(X_n - a)^-] \leq \mathbb{E}[|X_n|] + |a|. \end{align*} The second inequality uses $(X_n - a)^- = \max(a - X_n, 0) \leq |X_n - a| \leq |X_n| + |a|$. Since $N_n([a, b], X)$ is non-negative and increasing in $n$ with limit $N([a, b], X) = \lim_{n \to \infty} N_n([a, b], X)$, the [Monotone Convergence Theorem](/theorems/509) gives: \begin{align*} (b - a) \, \mathbb{E}[N([a, b], X)] = \lim_{n \to \infty} (b - a) \, \mathbb{E}[N_n([a, b], X)] \leq \sup_{n \geq 0} \bigl(\mathbb{E}[|X_n|] + |a|\bigr) < \infty. \end{align*} Therefore $N([a, b], X) < \infty$ a.s. for each rational pair $a < b$. [/step] [step:Apply the upcrossing convergence criterion on a countable intersection] The [set](/page/Set) of rational pairs $\{(a, b) \in \mathbb{Q}^2 : a < b\}$ is countable. For each such pair, define $\Omega_{a,b} = \{N([a, b], X) < \infty\}$, which has $\mathbb{P}(\Omega_{a,b}) = 1$ by the preceding step. Taking the countable intersection: \begin{align*} \mathbb{P}\Bigl(\bigcap_{a < b \in \mathbb{Q}} \Omega_{a,b}\Bigr) = 1. \end{align*} On the event $\Omega_0 = \bigcap_{a < b \in \mathbb{Q}} \Omega_{a,b}$, the [Convergence Criterion via Upcrossings](/theorems/1155) guarantees that $X_n(\omega)$ converges in $\overline{\mathbb{R}}$ for every $\omega \in \Omega_0$. Define $X_\infty(\omega) = \lim_{n \to \infty} X_n(\omega)$ for $\omega \in \Omega_0$ and $X_\infty(\omega) = 0$ for $\omega \notin \Omega_0$. Then $X_\infty$ is $\mathcal{F}_\infty$-measurable as a limit of $\mathcal{F}_n$-[measurable functions](/page/Measurable%20Functions). [/step] [step:Verify $X_\infty \in L^1$ and the limit is a.s. finite via Fatou's lemma] Since $|X_n| \to |X_\infty|$ a.s. on $\Omega_0$ and $|X_n| \geq 0$, [Fatou's Lemma](/theorems/510) gives: \begin{align*} \mathbb{E}[|X_\infty|] = \mathbb{E}\bigl[\liminf_{n \to \infty} |X_n|\bigr] \leq \liminf_{n \to \infty} \mathbb{E}[|X_n|] \leq \sup_{n \geq 0} \mathbb{E}[|X_n|] < \infty. \end{align*} In particular, $X_\infty$ is a.s. finite (an integrable [function](/page/Function) is finite a.s.), so the convergence $X_n \to X_\infty$ is in $\mathbb{R}$, not $\pm \infty$. For the final assertion: if $X$ is a non-negative supermartingale, then $\mathbb{E}[|X_n|] = \mathbb{E}[X_n] \leq \mathbb{E}[X_0] < \infty$ (by the supermartingale inequality), so the $L^1$-boundedness condition is automatically satisfied. [guided] The logical flow is: $L^1$-bounded $\Rightarrow$ finite expected upcrossings (via Doob's inequality) $\Rightarrow$ a.s. finite upcrossings (via MCT) $\Rightarrow$ a.s. convergence in $\overline{\mathbb{R}}$ (via the upcrossing criterion) $\Rightarrow$ a.s. convergence in $\mathbb{R}$ (via Fatou). The step from "a.s. finite upcrossings for each rational pair" to "a.s. convergence" requires a countable intersection argument. This is where the restriction to *rational* pairs $a < b$ is essential: we need a countable collection to take the intersection and preserve the probability-one event. The upcrossing criterion ([Convergence Criterion via Upcrossings](/theorems/1155)) ensures that rational pairs suffice: if $\liminf < \limsup$, there exist *rational* $a, b$ between them (by density of $\mathbb{Q}$). The Fatou step is not just a technicality — it eliminates the possibility that $X_n \to \pm \infty$. Without $L^1$-boundedness, a supermartingale could diverge to $-\infty$ (consider $X_n = -n$, a supermartingale that is not $L^1$-bounded since $\mathbb{E}[|X_n|] = n \to \infty$). [/guided] [/step]