Stationarity of a process

Question

I have a process: $$\epsilon_t=\nu_t\sqrt{\alpha_0+\alpha_1\epsilon^2_{t-1}}$$ where $\nu_t$ is a white noise with variance equal to $1$. My textbook says that the conditional variance of $\epsilon_t$ is not constant, but what can I say about the unconditional variance of $\epsilon_t$? It is said to me that $E[\epsilon^2_t]=E[\epsilon^2_{t-1}]$ (i.e. $\epsilon_t$ is a stationary process) but I am not able to prove it.