In Mathematical Statistics: Basic Ideas and Selected Topics (Kjell A. Doksum, Peter J. Bickel), problem 2.2.29 (a) asks:
Let $\epsilon_1,...,\epsilon_{n+1}$ be i.i.d. with mean zero and $Y_i = \mu + \epsilon_i/2 + \epsilon_{i+1}/2 $ for some fixed $\mu$ and $i=1,...,n+1$. Show that: $$\mathbb{E}[Y_{i+1} \mid Y_1, ..., Y_i] = \mu/2 + Y_i/2 $$
I can solve it once I know that $\mathbb{E}[Y_{i+1} \mid Y_1, ..., Y_i] = \mathbb{E}[Y_{i+1} \mid Y_i]$. But how do you show $\mathbb{E}[Y_{i+1} \mid Y_1, ..., Y_i] = \mathbb{E}[Y_{i+1} \mid Y_i]$ ?
Please prove it very formally and not with any "intuition". Thank you very much in advance!