Brownian Motion is Martingale

Question

I'm reading a book, and they say Brownian Motion is martingale then show it with the following calculation:

Suppose $(B_t)$ is brownian motion which generates the filtration $\mathcal F_t$ (for all $B_s$ such that $s \leq t$). Then we have: $$E[|B_t|]^2 \leq E[|B_t|^2] = |B_0|^2 + nt$$ and if $s \geq t$ then they do a calculation to show $E[B_s|\mathcal F_t] = B_t$

1. Why are they showing $s \leq t$ case? Isn't the defining property of martingale that in the $s \geq t$ case, $E[B_s|\mathcal F_t] = B_t$?
2. The steps to obtain the inequality are a bit unclear to me.

Answer 1

$\text{Var}(X) := E[(X-E(X))^2] = E(X^2)-E(X)^2$. From the definition, you easily see that the variance is positive, hence $E(X^2)\geq E(X)^2$.

Answer 2

1) They never showed $s \leq t$ case. If you’re refering to the definition of $\mathcal{F}_t$, that has nothing to do with the inequality. A stochastic process is defined with respect to a filtration.

2) $f(x) = x^2$ is convex so by Jensen’s inequality, $f(E(|B_t|)) \leq E(f(|B_t|))$