I'm trying the classic AirPassengers dataset in R and tried removing the seasonal component using deSeas <- forecast::seasadj(decompose(d,"multiplicative")), but it doesn't seem to go away.
Original data:

Seasonality viewed with plot(decompose(d,"multiplicative")):

Seasonality adjusted with

deSeas <- forecast::seasadj(decompose(d,"multiplicative"))
plot(deSeas)

But decompose still shows a seasonal pattern, plot(decompose(deSeas,"multiplicative")):

I don't understand why there still is a seasonality. Should I de-seasonalize it again? Will arima be able to re-incorporate the single de-seasonalized or double de-seasonalized data back into the predictions that it tries to make?