I am currently reading the famous almost sure blog by George Lowther about stochastic calculus. I am currently reading the section about the Burkholder-Davis-Gundi-Inequality (BDG inequality). At the end of the blog entry under the title "Continuous-time Local Martingales" he proves the BDG inequality for $p \geq 1$. He uses an approximating sequence
$$[X]^{(n)}_t = \sum_{k=1}^\infty(X_{t^n_k \land t}-X_{t^n_{k-1} \land t})^2 \rightarrow [X]_t$$
which converges in u.c.p. (i.e. uniformly over compact sets in probability) to the quadratic variation of the local-martingale $X$ and by passing to a subsequence we can assume convergence uniformly over compacts P-a.s.. The parts of the proof I do not understand, are where he defines
$$M_t := \sup_{n \in \mathbb{N}}[X]_t^{(n)}$$
and claims that $(M_t)_{t \in [0,\infty)}$ is RCLL (right continuous admitting left limits) and that its jumps are bounded in the following way $$\Delta M_t \leq \sup_{n \in \mathbb{N}} \Delta [X]_t^{(n)}$$
Has anyone read his blog and knows why we can conclude RCLL and such a bound for the jumps?
Thanks a lot in advance!