Let $M_t$ be an a.s. continuous local martingale and $M_0=0$.
Let $A_t$ be an adapted, a.s. continuous and increasing process,and $A_0=0$.
Let $\pi_n: 0 = t_0^n < . . . t_{k_n}^n=t$ be a sequence of partitions of $[0,t]$ with $|\pi_n| \to 0$. Then $$V_n(M):=\sum_{I=0}^{k_n-1}(M_{t_{i+1}^n} - M_{t_i}^n)^2 \to_{n\to \infty} A_t = \langle M\rangle_t$$ in probability.
In some proof I found for example that: $\sup(M_{t_{i+1}^n} - M_{t_i}^n) \to_{n \to \infty} 0$ a.s. (because of the continuity of M).
But, then, why also $V_n(M)$ does not go to $0$, when $n \to \infty$? It seems to me that by the continuity of M, $V_n(M)$ is again a sum of $( M_{t_{i+1}^n} - M_{t^n_i})$ that are $=0$.
Why this is not true?
It is because your number of summands is also increasing. A similar example would be this sum (or any Riemann integral)
$$ \sum_{i=1}^n \frac{1}{n} $$
Here, $\frac{1}{n} \rightarrow 0$ but you probably would not argue that this sum converges to 0.