It was probably not tarnished when you got it. Brass and copper come out of a box bright, and the change is slow enough that you do not see it happen. Then one day the lamp base is brown, and a cloth does nothing at all.

That brown is not a film that settled. It is the metal itself reacting with the air, a layer of oxide and sulphide grown out of the surface. This is worth knowing because it explains why washing achieves nothing: you cannot wipe off something that is chemically part of what it is sitting on. It has to be converted back or lifted off.

What a polish does is dissolve that layer and carry it into the cloth. You can watch it work, which is unusual. The cloth turns black. That black is your tarnish, and the fact that it is on the cloth is the proof it is no longer on the metal.

Which sets the technique. Work a small area, keep moving to an unused part of the cloth, and stop when the cloth stops turning colour. Do not press harder to speed it up. Pressure does nothing chemical, and the abrasive is mild for a reason.

Two honest limits. First, most decorative brass sold in the last forty years is brass plated over steel or zinc, and plating is thin. Polish will bring it up, and if you keep going you will go through it to the grey underneath, which is permanent. Test somewhere hidden. Second, if the piece has a factory lacquer over it, the tarnish is under the lacquer and no polish will reach it. Peeling, patchy shine is the sign. That is a stripping job first.