# is it possible a nonstationary time series, to produce a stationary ARMA model?

I Have a variable (time series) which is nonstationary. I found that from the graph which seems to have a stochastic trend and the correlogram has a typical nonstationary pattern. After that, I've been asked to find the parsimonious ARMA model (using information criteria), and by following the Box-Jenkins methodology, go down till forecasting. Notice that at this point, I havent been asked to make a unit root test to check if the series is stationary or not...

Is it possible a nonstationary series to produce a stationary ARMA model, which will give us reliable forecasting?

After the forecasting, they ask me to make a unit root test (where of course I find that the variable is non stationary). As a next step I take the variable in 1st differences, and follow the same procedure to find an ARMA model with best information criteria. I also make some comparissons with the model that I found at the 1st step, but after that they dont ask me to follow the Box-Jenkins methodology again and make a forecast from the ARMA model produced by the differenced variable...

So what I’m confused about is:

Why did they ask me to find an ARMA from a non stationary variable? Can a nonstat. Variable produce stationary ARMA model and reliable forecasting?

Wouldn’t it be normal to ask me to follow the Box Jenkins methodology in the second model (where the variable is differenced) and make a forecast from there?