Conditional expectation of a white noise process

[citizen333]

Hi all.

The problem is as follows:

Y_t = ε_t + θ₁ε_t−1 + θ₂ε_t−2ε_t ∼ WN(0, σ²)

White noise is in the weakest form: there is no Gaussian or even iid condition, only absence of autocorrelation.

We need to find one step ahead forecast of this process: E_t+1(Y_t) - ?
The problem I face is that E_t+1(ε_t) is unknown. Obviously it must be zero, however I don't know how to prove it. If anyone knows the proof please help.

Thanks in advance.