Martingales, finite expectation I have some uncertainties about one of the requirements for martingale, i.e. showing that $\mathbb{E}|X_n|<\infty,n=0,1,\dots$ when $(X_n,n\geq 0)$ is some stochastic process. In particularly, in some solutions I find that lets say $\mathbb{E}|X_n|<n$ is accepted, for example here (2nd slide, 1.2 example). So my question is: what is the way of thinking if n goes to infinity? Why are we accepting n as a boundary or maybe I misunderstood something? 

The condition $\mathbb E|X_n|\lt n$ is odd. What is required for $(X_n)$ to be a martingale is, in particular, that each $X_n$ is integrable (if only to be able to consider its conditional expectation), but nothing is required about the growth of $\mathbb E|X_n|$.
Consider for example a sequence $(Z_n)$ of i.i.d. centered $\pm1$ Bernoulli random variables, and a real valued sequence $(a_n)$. Then $X_n=\sum\limits_{k=1}^na_kZ_k$ defines a martingale $(X_n)$ and $\mathbb E|X_n|\gt |a_n|$ by convexity. One sees that the growth of $n\mapsto\mathbb E|X_n|$ cannot be limited a priori.
