Inverse-clock proof of the Dambis-Dubins-Schwarz theorem ($B_u=M_{T_u},M_t=B_{V_t}$)

For a continuous strictly increasing unbounded bracket $V$, the inverse times $T_u=\inf \{t:V_t>u\}$ satisfy $V_{T_u}=u$ and $T_{V_t}=t$. Optional sampling of the original martingale stopped at bounded bracket levels makes $M_{T_u}$ a continuous local martingale in ${\mathcal{F}}_{T_u}$. Time-changing the square-minus-bracket martingale gives its bracket $u$; the Lévy characterization of Brownian motion identifies it as Brownian.