Where is the purchase history on iPhone apps? How do I find out if I've already purchased an iPhone app?

I don't have a credit card so I'm worried about how I can pay to download apps if they're really not free. Even if I have chosen NONE in the billing method, they will still allow me to download paid apps even if I didn't select a payment method. 

I created an iTunes account to download free apps through my iPhone. When I was making the account, there was an option on how to pay the bills: 


choose a credit card
other means, or
simply choose the "NONE" 


I chose "None" because I thought that it would still prompt for credit card information if you were to purchase an app that needed payment. I only wanted to download free apps and that's also why I chose "none".

After a few days of downloading free apps, I clicked on updates and noticed one of them saying that since I already purchased it first, the update was at a discount.

I cancelled the update since I have no credit card.

Days later I noticed the same app in the store, but free. Trying to install it, it would not download and said there was an update pending on my previous version. I waited for a screen to appear about payment information, but none came.

I wanted to cancel paying for this update but haven't seen the option to do so.

I don't know if my wanton clicking means that I've paid for this app without realising it. I have my billing address on there, so maybe there will be a bill coming along.
You can see your purchase/download history in the App Store application on your device. It is in the Updates tab. This feature was introduced in iOS 4.3.3.