In my exam on probability theory I got a question which I think probably contains an error since I simply cannot prove that $X_n$ is adapted. Moreover I cant even prove the martingale property. I also tried to modify the question tinkering with $X_n$ and the sigma fields to fix the problem so that I could make it a martingale but I was unsuccessful. Can somebody help! Is it really a martingale? And if not how can I modify the definition of random variables to make it a martingale. Clearly the expectations are finite and equal to $1$ in this setting but other than that I cant prove anything.
The set $\left[n+1,\infty\right)$ belongs to $\mathcal F_n$, since it can be written as $\left[n,\infty\right)\setminus \left\{n\right\}$ (the $\sigma$-algebras generated by $\left\{\left\{1\right\},\left\{2\right\},\dots,\left\{n\right\},\left[n+1,\infty\right)\right\}$ or $\left\{\left\{1\right\},\left\{2\right\},\dots,\left\{n\right\},\left[n,\infty\right)\right\}$ are the same, but the later is not a partition. Therefore, $X_n$ is $\mathcal F_n$-measurable.
Since $$ X_n=\left(n+1\right)\mathbf 1_{\left[n+1,\infty\right)} =\left(n+1\right)\mathbf 1_{\left[n,\infty\right)}-(n+1)\mathbf 1_{\left\{n\right\}}=X_{n-1}+\mathbf 1_{\left[n,\infty\right)}-(n+1)\mathbf 1_{\left\{n\right\}}, $$ it suffices to prove that $$ \mathbb E\left[\mathbf 1_{\left[n,\infty\right)}-(n+1)\mathbf 1_{\left\{n\right\}}\mid\mathcal F_{n-1}\right]=0. $$