I was reading a proof in Kallenberg's book foundations of modern probability but there are two things I don't understand.
Notation : For continuous local martingales $M,N$, the quantity $[M,N]$ denotes the covariation process and we denote $[M]:=[M,M]$.
Proposition 18.5 For any continuous local martingales $M_n$ starting at $0$, we have $$\sup\limits_tM_n(t)\overset{P}{\to}0\iff[M_n]_{\infty}\overset{P}\to 0.$$
proof : First let $\sup\limits_{t}M_n\overset{P}{\to}0$. Fix any $\epsilon>0$, and define $\tau_n=\{t\ge 0;|M_n|>\epsilon\},n\in\mathbb{N}$. Write $N_n=M_n^2-[M_n]$, and note that $N_n^{\tau_n}$ is a true martingale on $\overline{\mathbb{R}}_+$. In particular $E[M_n]_{\tau_n}\le \epsilon^2$, and so by Chebyshev's inequality $$P\{[M_n]_{\infty}>\epsilon\}\le P\{\tau_n<\infty\}+\epsilon^{-1}E[M_n]_{\tau_n}\le P\{\sup\limits_t M_n>\epsilon\}+\epsilon.$$ [...]
The first thing I don't understand is how we can say $E[M_n]_{\tau_n}\le \epsilon^2$.
The second thing that is not clear to me is in the step
$P\{[M_n]_{\infty}>\epsilon\}\le P\{\tau_n<\infty\}+\epsilon^{-1}E[M_n]_{\tau_n}$
It seems that there is an intermediate step $P\{[M_n]_{\infty}>\epsilon\}\le P\{\tau_n<\infty\}+P\{[M_n]_{\tau_n}>\epsilon\}$ which I don't understand (and then we apply Markov on the right hand side).
Since $M^{\tau_n}_n$ is bounded, $N^{\tau_n}_n$ is a uniformly integrable martingale, hence by Doob's stopping theorem, $$ E N_{\tau_n}=0, $$ or equivalently, $$ E[M_n]_{\tau_n}=E\vert(M_n)_{\tau_n}^2\vert\le E\varepsilon^2=\varepsilon^2. $$
For the second thing, $$ P\{[M_n]_\infty>\varepsilon\}=P\{[M_n]_\infty>\varepsilon,\tau_n<\infty\}+P\{[M_n]_\infty>\varepsilon,\tau_n=\infty\}. $$
On the one hand, $$ P\{[M_n]_\infty>\varepsilon,\tau_n<\infty\}\le P\{\tau_n<\infty\}. $$
On the second hand, using Markov's inequality for the last inequality, $$ P\{[M_n]_\infty>\varepsilon,\tau_n=\infty\}=P\{[M_n]_{\tau_n}>\varepsilon,\tau_n=\infty\}\le P\{[M_n]_{\tau_n}>\varepsilon\}\le\frac{E[M_n]_{\tau_n}}{\varepsilon}. $$