Prove that a stochastic process is a Martingale

Question

Let $\Omega=\mathbb{N}\setminus\{0\}$ , $\mathbb{P}(A)=\sum_{k\in A}2^{-k}$.
Let's consider the stochastic process $M_n=\omega$ $\text{ }$ if$\text{ }$ $\omega\leq n$, $M_n=n+2$ $\text{ }$if$\text{ }$ $ \omega > n$.
How can I prove that $\mathbb{E}[M_{n+1}|\mathfrak{F}_n]=M_n$?

I tried to calculate $\mathbb{E}[M_{n+1}-M_n|\mathfrak{F}_n]=0$. The result I got was: $$-1*\frac{1}{2^{n+1}}+\sum_{k=n+2}^\infty \frac{1}{2^k}=0$$ that should be enough to prove what I wanted, but I don't know if it can be done this way.


Edit: the solution I proposed is correct if anyone is interested.
Answer to the comment: I considered $\mathfrak{F}_n$ to be $\{\{1\},\{2\},...,\{n\},\{n+1,n+2,.....\}\}$.