Let $Z$ be a random variable and $(F_n)_{n \ge1}$ a filtration, then $(X_n)_{n\ge1}$ where $$ X_n:= \mathbb{E}[Z|F_n]$$ is a martingale.
where: $$Z = \sum_{k \ge1} \frac{Y_k}{2^k}$$ where $Y_1, Y_2, ...$ are independent Bernoulli trials $(1/2)$.
I now would like to prove that the sequence of random variables $(X_n)_{n\ge1}$ is a martingale. Then I have to prove that: $\mathbb{E}[|X_n|] < \infty$ and $\mathbb{E}[X_{n+1}|F_n] = X_n$
I would appreciate any hint on where to start!
For any random variable $Z$ with finite expectation $(E(Z|\mathcal F_n))$ is a martingale. This follows immediately from definition of a martingale and the fact that $E(E(Z|\mathcal G_1)|\mathcal G_2)=E(Z|\mathcal G_2)$ if $\mathcal G_2 \subseteq \mathcal G_1$. In this case $EZ=\sum_k \frac 1 {2^{k+1}} <\infty$. [Note that the expectation and the sum over $k$ can be switched because all the terms are non-negative].