Define:
For $n \geq 0$, on note $X_n=(n+1) \mathbb{1}_{[n+1,+\infty}$, and $\mathcal{F}_n=\sigma(\{1\},\{2\}, \ldots,\{n\},[n+$ $1,+\infty[)$
and $\forall k \in \mathbb{N}^*, \mathbb{P}(\{k\})=\frac{1}{k}-\frac{1}{k+1}$
I am trying to prove that it's a martingale.
We have that $$X_{n+1}=(n+2) 1_{[n+2,+\infty}[$$
$$\mathbb{E}\left[X_{n+1} \mid \mathcal{F}_n\right]=(n+2) \mathbb{E}\left[1_{[n+2,+\infty[} \mid \mathcal{F}_n\right]$$
And because $\{k\}$ for $k \geq n+2$ is independant from $\mathcal{F}_n$
$$\mathbb{E}\left[1_{[n+2,+\infty} \mid \mathcal{F}_n\right]=\mathbb{P}([n+2,+\infty[)$$
$$\mathbb{P}\([n+2,+\infty[)=\sum_{k=n+2}^{\infty}\left(\frac{1}{k}-\frac{1}{k+1}\right)$$
$$=\frac{1}{n+2}-\frac{1}{n+3}+\frac{1}{n+3}-\frac{1}{n+4}+\ldots$$
$$=\frac{1}{n+2}$$
$$\mathbb{E}\left[X_{n+1} \mid \mathcal{F}_n\right]=(n+2) \times \frac{1}{n+2}=1$$
But this is supposed to be equal to $X_n$. What am I doing wrong ?
$\{k\}$ is not independent of $\mathcal F_n$.
Since $\mathcal F_n$ is generated by a partition, $$ \mathbb E\left[X_{n+1}\mid\mathcal F_n\right]=\sum_{k=1}^n \mathbb E\left[X_{n+1}\mathbf{1}_{\{k\}}\right]\frac 1{\mathbb P(\{k\})}\mathbf{1}_{\{k\}}+\mathbb E\left[X_{n+1}\mathbf{1}_{[n+1,\infty)}\right]\frac 1{\mathbb P([n+1,\infty))}\mathbf{1}_{[n+1,\infty)}. $$ Only the last term remains and since $[n+2,\infty)\subset [n+1,\infty)$, we have $$ \mathbb E\left[X_{n+1}\mathbf{1}_{[n+1,\infty)}\right]\frac 1{\mathbb P([n+1,\infty))}=(n+2)\frac{\mathbb P([n+2,\infty))}{\mathbb P([n+1,\infty))}, $$ which, after simplification, gives the result.