I'm having a little difficulty proving the corresponding forecast errors for an AR(1) model with a horizon of h. In other words, I want to show that $V_{T+h|T} = \epsilon_{T+h}+\beta \epsilon_{T+h-1}...\beta^{h-1}\epsilon_{T+1}$ given AR model $y_T = \beta y_{T-1} + \epsilon_{T}$
I tried breaking this down by doing the 1 step ahead and 2 step ahead as follows:
$V_{T+1|T} = (y_{T+1} - y_{T+1,T}) = \beta y_t + \epsilon_{T+1} - \beta y_t = \epsilon_{T+1}$
$V_{T+2|T} = (y_{T+2} - y_{T+1,2}) = (\beta y_{T+1} + \epsilon_{T+2}) - \beta^2y_{T+1} = \beta(y_{T+1} - \beta y_{T+1}) + \epsilon_{T+2} = \beta \epsilon_{T+1} + \epsilon_{T+2}$
And using this intuition, I tried the following attempt for h:
$V_{T+H|T} = (y_{T+H} - y_{T+H,T}) = \beta y_{T+h-1} + \epsilon_{h+T} - \beta^h {y_t} = \beta(y_{T+h-1} - \beta^{h-1}y_T) + \epsilon_{T+h} = \beta(\beta^{h-1} \epsilon_{T+h-1} + \epsilon_{T+h}$