Suppose that $\nu$ is a caglad process. In a paper, it is claimed that $$ \lim_{n\rightarrow\infty}n^{2\,r}\,\sum_{j=1}^{n}\left(\left(\int_{j/n}^{(j+1)\,/n}\nu_u^2\,du\right)^{r}-\left(\int_{(j-1)/n}^{j/n}\nu_u^2\,du\right)^{r}\right)^2=\lim_{n\rightarrow\infty}\sum_{j=1}^{n}\left(\nu_{j/n}^{2\,r}-\nu_{(j-1)/n}^{2\,r}\right)^2=[\nu^{2\,r}]_{1} $$ where the limits are intended in probability and where $[\nu^{2\,r}]_{1}$ denotes the quadratic variation of $\nu^{2\,r}$ in $\left[0,1\right]$.
For me it is clear how to prove the results assuming that $$ \left|\nu^{2}_t-\nu^{2}_s\right|\leq \left|t-s\right|, $$ however, I would be better to have the proof by just assuming left-continuity.