Does my bathroom mirror-light work?

I have a very basic understanding of voltage/current/resistance but have always wanted to know more. I did a year's physics about 10 years ago and never anything since. I can work out how much it costs to power
my TV for an hour though (which I was quite proud of)!
Anyway, now I've moved into a new flat and the bathroom mirror light doesn't work so I thought I'd do some digging. Bought a bulb and put it in, no joy. Bought a second as I though I may have broken the filament on the first but no joy either. So I borrowed an ammeter. Tested the resistance across the long filament bulb and it's about what you get from a bulb I know works (albeit an ordinary bulb). So I'm guessing the bulb is Ok.
Then I removed the little panel on the back of the bathroom mirror light and I can see where it's connected to the house supply. I can test the voltage there by touching each of the contacts with the two wires on the meter and I get 237V (in Ireland that's good). Then I put the bulb between the two metal contact points (carefully as I don't know if the light is on or off as there is only a pull-cord switch) and test the two ends of the bulb but I get nothing.
My question: Should I remove the bulb and put the ammeter in it's place testing for AMPS in A/C mode? Will this blow my Ammeter if there is a current? It has 2AMP and 10AMP fused connectors, which should I use?
My goal: I want to know if the light is broken. I want know which is the on/off position. Am I right in saying that if there's a resistance a cross the bulb the filament must be Ok?
Thanks so much in advance for any and all help,
Kind regards, Andoni.
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