I want to show that for any two continuous local martingales and any stopping time $T$, $\langle M^T, N \rangle = \langle M, N \rangle ^T$.
The method suggested in Revuz and Yor as an exercise is to show that $M^T(N^T-N)$ is a local martingale so that trivially $$M^TN^T - \langle M, N \rangle^T - M^T(N^T-N) = M^TN - \langle M, N \rangle^T$$ is a local martingale and we may conclude by the uniqueness of the quadratic covariation process.
I cannot show that $M^T(N^T - N)$ is a continuous local martingale. Any suggestions on how to do this?