= Truncated layer-cake proof of the L2 maximal inequality
{title2=$\|M_t\|_2\leq2\|X_t\|_2$}
For a nonnegative <submartingale> with maximum $M_t$, the first-moment maximal estimate is $\lambda\mathbb P(M_t\geq\lambda)\leq\mathbb E[X_t\mathbf1_{\{M_t\geq\lambda\}}]$. Integrate $2\lambda$ times the tail probability only up to $R$, obtaining $\mathbb E(M_t\wedge R)^2\leq2\mathbb E[X_t(M_t\wedge R)]$. The <Cauchy-Schwarz inequality> gives $\|M_t\wedge R\|_2\leq2\|X_t\|_2$, and <monotone convergence> gives the result as $R\to\infty$. Truncation avoids assuming beforehand that the maximum has a finite second moment.
// Target: brownian-motion.bigb
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