First order variation of the predictable quadratic variation

In the book "Brownian Motion and Stochastic Calculus" by Karatzas and Shreve it is said that if $$X\in\mathcal{M}_2^c$$ (i.e. $$X$$ is a continuous square integrable martingale) the predictable quadratic variation $$\left_t$$ (defined, through the Doob-Meyer decomposition, as the a.s. unique natural increasing process $$Y$$ such that $$X^2-Y$$ is a martingale) is its own first variation, that is

$$p-\lim_{n\rightarrow\infty}\sum_{k=1}^n\left| Y_{t_{k}}-Y_{t_{k-1}} \right|=\left_t$$

where $$0=t_1 is a partition of $$[0,t]$$. Although this should be obvious, I can't figure out why the identity holds.

Since $$Y$$ is increasing, $$|Y_{t_k} - Y_{t_{k-1}}| = Y_{t_k}-Y_{t_{k-1}}$$, so this is a telescoping sum: \begin{align*} \sum_{k=1}^n |Y_{t_k} - Y_{t_{k-1}}| &= \sum_{k=1}^n (Y_{t_k}-Y_{t_{k-1}}) \\ &= Y_{t_n} - Y_{t_1} \\ &= Y_t = \langle X\rangle_t. \end{align*} The fact that $$Y_{t_1}=Y_0 = 0$$ is because part of the definition of the quadratic variation is $$\langle X\rangle_0 = 0$$.