I am faced with the following exercise:
Let $X_{1},X_{2},...$ be independent random variables satisfying $\mathbb{E}(X_{n}^{2})<\infty$ and $\mathbb{E}(X_{n})=0$ for all $n\in\mathbb{N}_{0}$. Define the martingale $M$ by $M_{n}=\sum_{k=1}^{n}X_{k}$. Determine $\langle M\rangle$, the compensator of the quadratic variance of $M$.
By a theorem in my lecture notes, I have that $\Delta\langle M\rangle_{n}=\mathbb{E}[(M_{n}-M_{n-1})^{2}|\mathcal{F}_{n-1}]$ holds almost surely. Assuming $\langle M\rangle_{0}=0$, this would simply give
$\langle M \rangle_{n}=\sum_{k=1}^{n}[X_{k}^{2}|\mathcal{F}_{n-1}]$
Because of the simplicity of this solution, I feel as though I am missing some bigger picture. Is this approach (appart from the lack of rigour) correct?