This seems like a simple question, but I am stumped. I know the proofs for quadratic variation and cross variation, etc..., but for some reason can't understand why the following doesn't make sense to do: \begin{equation} [Z,Z]([0,t])=\lim_{\delta_n\to 0}\sum_{i=1}^{n}(\,Z(t_i)-Z(t_{i-1})\,)^2 \leq \lim_{\delta_n\to 0}\sum_{i=1}^{n}(\max_i [Z(t_i)-Z(t_{i-1})]\,)^2 = 0 \\\implies \lim_{\delta_n\to 0}\sum_{i=1}^{n}(\,Z(t_i)-Z(t_{i-1})\,)^2 = 0 \end{equation}
I am unsure why this reasoning can't hold, and it's making the proofs for QVs of other processes confusing. It seems like I have passed the limit through the summation and then assumed the product of the maxes is $0$, so if those don't hold I am wondering why not. It seems like passing a limit that is essentially $\lim_{n \to \infty}$ through the sum doesn't make sense to do, so would that be the mistake here? I was thinking of it in the sense of: \begin{equation} \lim_{n \to \infty} \sum_{i=1}^nx_i(n) = \lim_{n \to \infty} \sum_{i=1}^n\lim_{n \to \infty}x_i(n) \end{equation} or \begin{equation} \lim_{n \to \infty} \sum_{i=1}^nx_i(n) = \sum_{i=1}^{\infty}\lim_{n \to \infty}x_i(n) \end{equation} Thanks for the help!
Note that the $n$ in the upper bound of the sum depends on the partition and hence depends on your $\delta_n$. In fact $n \to \infty$ as $\delta_n \to 0$ (to chop up $[0,t]$ in to blocks of size $\delta_n$ you need more and more blocks as $\delta_n$ gets smaller). This means that you can't pass the limit through the sum because the upper bound of the sum depends on the thing you take the limit with respect to.
What is happening is that whilst $(\max_i [Z(t_i)-Z(t_{i-1})]\,)^2$ gets small when $\delta_n$ gets small, the number of such terms contributing to the sum gets larger and these two effects push the opposite way which prevents necessary convergence to $0$.