How To Take a City-State?!?!?!

[arclight]

Okay, I've been playing Civ since Civ2 and I'm pretty lost here on my first Civ5 game.

Am I right in saying that I can range attack a city-state all day and night and it will never, ever, ever go below a certain threshold of strength? I have been repeatedly pounding a city-state via ranged attacks for, well, two hours...and in previous editions of the game, a city would be pretty quickly weakened so that you can just walk right in. However, no matter how much I pound it, it always is able to destroy my incoming melee units.

What am I missing? How do you actually take one of these?

Thanks!