Let $\{X_n\}_{n\in\mathbb{N}}$ be a process with independent increments. We define $Y_n = X_n−E[X_n]$. Prove that $\{Y_n\}_{n\in\mathbb{N}}$ is a martingale with respect to $\{X_n\}_{n\in\mathbb{N}}$.
I am trying to prove that: $E[Y_{n+1}|X_0,\ldots,X_{n}] = X_n$ then \begin{align} E[Y_{n+1}|X_0,\ldots,X_{n}] &= E[X_{n+1}-E[X_{n+1}]|X_0,\ldots,X_{n}]\\ &=E[X_{n+1}|X_0,\ldots,X_n] - E[X_{n+1}]\\ &=E[X_{n+1}-X_n +X_n|X_0,\ldots,X_n] - E[X_{n+1}]\\ &= E[X_{n+1}-X_n|X_0,\ldots,X_n] + E[X_{n}|X_0,\ldots,X_n]- E[X_{n+1}]\\ &= E[X_{n+1}-X_n] + X_n -E[X_{n+1}]\\ &= Y_n \end{align} The procedure correct? and why is $E[X_{n}|X_0,\ldots,X_n] = X_n$ true?