Is there any device you can tell if borders will go away?
I can't guarantee anything, but the second device you showed (the one I said should work, not the one scarmullet quoted) is pretty good bet.
So your TV does not have zoom button? that should remove the borders without any extra hardware.
If no Zoom button, do consider is it cheaper / easier just to buy new TV.
HDTVs have come down in price big time.