I quote Jacod-Protter
Given a probability space $\left(\Omega,\mathcal{F},\mathbb{P}\right)$ and an increasing sequence of $\sigma$-algebras $\left(\mathcal{F}_n\right)_{n\geq0}$, let $M=\left(M_n\right)_{n\geq0}$ be a sequence of integrable r.v.'s, each $M_n$ being $F_n$-measurable and $M_n^{*}=\sup_{j\leq n} |M_j|$
At this point it is stated that:
Note that ($M_n^{*}$) is an increasing process and a submartingale, since \begin{equation} \mathbb{E}[{M_n^{*}]} \leq \mathbb{E}\left[\sum_{j=1}^{n}|M_j|\right]<\infty \end{equation}
that is, $\mathbb{E}[{M_n^{*}]} \leq \mathbb{E}\left[\sum_{j=1}^{n}|M_j|\right]<\infty$ implies that ($M_n^{*}$) is an increasing process (and so a submartingale as well, since an increasing sequence of integrable r.v.'s, each being $\mathcal{F}_n$-measurable, is a submartingale, and I know how to prove this).
However, my question is: why in the above text is it stated that $\mathbb{E}[{M_n^{*}]} \leq \mathbb{E}\left[\sum_{j=1}^{n}|M_j|\right]<\infty$ implies that ($M_n^{*}$) is an increasing process (and so a submartingale as well)?
In which aspect does the inequality $\mathbb{E}[{M_n^{*}]} \leq \mathbb{E}\left[\sum_{j=1}^{n}|M_j|\right]<\infty$ (which is clearly correct in its formulation) tell me that ($M_n^{*}$) is an increasing process? Would not it be more correct to simply state that, by definition of $\sup$ function, for each $m<n$: \begin{equation} M_n^{*}=\sup_{j\leq n}|M_j|> M_m^{*}=\sup_{j\leq m}|M_j| \end{equation} which means, by definition, that ($M_n^{*}$) is an increasing sequence?
$(M_n^*)_n$ is clearly an increasing process since by definition it is a running maximum. The inequality $\mathbb{E}[M_n^*] \leq \mathbb{E}\sum_{j=1}^n |M_j|$ is used to prove that $M_n^*$ is integrable, which is needed to prove that it's a submartingale.