= Solution
Let $X,Y$ be two continuous solutions on the same <filtered probability space>, driven by the same <Brownian motion> and with $X_0=Y_0$ <almost surely>. Write $D=X-Y$ and choose a common <Lipschitz constant> $L$ for $b,\sigma$. Introduce the <stopping time> $\tau_n=\inf\{t\geq0:|X_t|+|Y_t|\geq n\}$. On $\{\tau_n=0\}$ the stopped difference is zero; before $\tau_n$, both coefficients and the difference are bounded.
The <Itô formula> and the zero <expectation> of the stopped <square-integrable> <Itô integral> yield
$$
\begin{aligned}
\mathbb E D_{t\wedge\tau_n}^2
&=\mathbb E\int_0^t\mathbf1_{\{s<\tau_n\}}\left(2D_s(b(X_s)-b(Y_s))+(\sigma(X_s)-\sigma(Y_s))^2\right)ds\\
&\leq(2L+L^2)\int_0^t\mathbb E D_{s\wedge\tau_n}^2\,ds.
\end{aligned}
$$
The <Gronwall inequality> implies $\mathbb ED_{t\wedge\tau_n}^2=0$. Continuous solution paths are bounded on each compact time interval, so $\tau_n\uparrow\infty$ <almost surely>. Letting $n\to\infty$ proves $X_t=Y_t$ <almost surely> at each fixed time, and a countable dense set of times together with continuity gives <indistinguishability of stochastic processes>. Therefore \b[the <stochastic differential equation> has <pathwise uniqueness>.]
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