proof of martingale convergence theorem

Let ${(X_n)}_{n\in \mathbb{N}}$ be a supermartingale such that $𝔼|X_n|\le M$, and let . We define a random variable counting how many times the process crosses the stripe between $a$ and $b$:

Obviously $U_{n+1}\ge U_n$ therefore $U_{\mathrm{\infty }}={lim}_{n\to \mathrm{\infty }}{U}_n$ exists almost surely. Next we will construct a new process that mirrors the movement of $X_n$ but only if the original process is underway of going from below $a$ to over $b$, and is constant otherwise. To do this let , for $k\ge 2$, and define $Y_0:=0$, $Y_n:=\sum _{k=1}^n{C}_k(X_k-X_{k-1})$. Then $Y_n$ is also a supermartingale, and the inequality $Y_n\ge (b-a)U_n-|X_n-a|$ holds, which gives $0\ge 𝔼(Y_n)\ge (b-a)𝔼(U_n)-𝔼|X_n-a|$. After rearrangement we get

$$𝔼(U_n)\le \frac{𝔼|X_n-a|}{b-a}\le \frac{𝔼|X_n|+|a|}{b-a}\le \frac{M+|a|}{b-a}.$$

Therefore by the monotone convergence theorem

which means $\mathbb{P}(U_{\mathrm{\infty }}=\mathrm{\infty })=0$. Since $a$ and $b$ were arbitrary $X={lim}_{n\to \mathrm{\infty }}{X}_n$ exists almost surely. Now the Fatou lemma gives

Thus $X_n$ is in fact convergent almost surely, and $X\in L^1.\mathit{∎}$