Why is any positive martingale the exponential of an Ito integral w.r.t. Brownian motion?
Here is a little proof.
For any positive P-martingale M, $dM_t = Mt ·1/M_tdM_t$. By Martingale Representation Theorem, $dMt = \Gamma_tdW(t)$ for some adapted process $\Gamma_t$. So $dM_t = M_t(\Gamma_t/M_t)dW$, i.e. any positive martingale must be the exponential of an integral w.r.t. Brownian motion.
Where in this proof are we using the fact that $M$ is positive?