In a proof that I'm reading I notice the following: Assume $M$ is a positive martingale with $E[M_t^2]= \frac1t$ then, by Jensen inequality: $$0\le \big(E[M_t]\big)^2\le E[M_t^2]= \frac1t\to0$$ when $t \to \infty$. This implies that $E[M_t]=0$ for all $t\ge1$
My question is: where did I use the assumption that M is a martingale. I guess that is the fact that a martingale has equal expectation for each $t$ . But still is not clear to me.
My interpretation is: $E[M_\infty]^2=0$ imply $E[M_\infty]=0 $ and then $E[M_t]=0$ for each $t$? However, doesn't I need to use some convergence theorem for obtain this result?
Is it true that for a martingale $E[M_\infty]=E[M_t]$?
For any martingale $M,$ we have for $s \leq t$
\begin{align*} E|M_t| &= E[E|M_t||\mathscr{F}_s] \tag*{(tower property)} \\ &\geq E|E [ M_t| \mathscr{F}_s]| \tag*{(Jensen's inequality)} \\ &= E|M_s|. \tag*{(M is a martingale)} \end{align*}
This means that $\sup_{t \geq 0} E|M_t| = \lim_{t \rightarrow \infty} E|M_t|.$
Also, from your question we know
$$0\leq \big(E[M_t]\big)^2\le E[M_t^2]= \frac1t,$$ Hence, $0 \leq E|M_t| < \frac{1}{t^2},$ and $$\sup_{t \geq 0} E|M_t|=\lim_{t \rightarrow \infty} E|M_t| = \lim_{t \rightarrow \infty} \frac{1}{t^2} = 0.$$ Therefore, $E|M_t| = 0$ for all $t$.