L2-bounded martingale convergence theorem (${\sup }_t\mathbb{E}{M}_t^2<\infty ⟹M_t\to M_{\infty }\text{ in }{L}^2$)

An $L^2$-bounded real martingale has a square-integrable terminal value, converges to it almost surely and in $L^2$, and satisfies $M_t=\mathbb{E}[M_{\infty }\mid {\mathcal{F}}_t]$. The $L^2$ Cauchy property follows from orthogonality of increments and monotonicity of $\mathbb{E}{M}_t^2$. This terminal representation permits optional sampling at finite unbounded stopping times and uniform-integrability control of stopped squares by conditional Jensen.