One way to prove that the starter solenoid is bad is to hit the starter while the key is turned to the start position. This can be a bit dangerous, so be careful. Grab a hammer or a sturdy piece of wood. Locate the starter on your vehicle, near the front of the transmission. On some vehicles it will be very hard to reach. Have a friend hold the key in the start position with their foot on the brake (and clutch, if your vehicle has a manual transmission). Give the starter a smack on the side or end, while being sure not to damage or dent it. If the starter runs after you hit it, you know you've got a bad solenoid and that you should replace the starter. If it doesn't run, you could still have a bad starter, but it could also be something else.
Hope this helps, and remember to be CAREFUL!