$\xi_1, .. \xi_n, ...$ - independent similarly distributed random variables with symmetric distribution. For any natural $n \ F_n= \sigma(\xi_1, ...\xi_n)$ and $S_n = \xi_1 + ... + \xi_n, \ S_0 = 0$. Prove, that for any $a$ sequence $X_k=P(S_{n-k}+S_k \leq a)$ is a martingale. ($k \in \{0, 1, ..., n\}$)
So we need to prove that $E(X_{k+1}|F_k)=X_k$
$X_k=P(S_{n-k}+S_k \leq a)=P(\xi_1+ ... + \xi_{n-k}+\xi_1+...+\xi_k \leq a)$
$X_{k+1}=P(S_{n-k-1}+S_{k+1} \leq a)=P(\xi_1+ ...+\xi_{n-k-1}+\xi_1+...+\xi_{k+1} \leq a) = P(\xi_1+...+\xi_{n-k}+\xi_1+...+\xi_{k}-\xi_{n-k}+\xi_{k+1} \leq a)$
In last sentence I expressed part that similar to $X_k$ but I don't know what to do with probability... Does anyone know what to do next?