While it might be unusual for all three lightbulbs to die at the same time, it is possible...did you change a bulb?
You might be able to visually detect an open circuit (broken wire, loose wire nut, etc.) if you took the lamp down. Without a multimeter and knowledge of how to use it, there isn't much else you can do.
When dealing with potentiall hot electrical wiring, it can be lethal, so know what you are doing or you can get hurt or killed.
If you want to try to look at things safely, turn off the circuit breaker or pull the fuse (if it is a really old panel!) before opening things up.
Are you sure you have voltage to the switch? The problem may not even be there, it could be further back towards the supply.
You really need some test equipment, a multimeter, or possibly a test lamp, or voltage detector. None of those are particularly expensive.