Could Someone Rate My First Approach?

Hello, all, I am wondering if someone can rate the first approach in my life. I am 23, brand new to dating and I have literally just made my first approach today. I was working out at the gym when I decided to go to the water fountain and I saw this good looking blonde sitting down while texting and talking to her friend. Right away I took off my head phones, sat beside her, and started a conversation like this (with a smile):

Me: Hello, how are you doing today?
Her: I'm doing good, thank you.
M: My name is Joshua, what's yours? (right away she smiles)
H: Brittany.
M: I'm new to this gym, are you? (right away she starts laughing)
H: No *laughs*
M: Before I continue, are you single?
H: *laughs* I'm sorry.
M: Sorry that you're taken?
H: Yes, I'm taken. (Her friend, who was sitting next to her, said that she was indeed taken and isn't lying)
Her Friend: You're funny, Brittany. *laughs*
M: Alright, I won't bother you anymore. (As I walk away to continue my workout I can hear her and her friend laughing and whispering something to each other)

I'm guessing that my mistake was asking her if she was single too quickly; but honestly I didn't feel like talking to her too much if it would have lead to no where. I'm just glad that she didn't say anything mean like they do in the movies. Can someone pleases rate my approach?

EDIT: I'm not butt-hurt by the experience at all. I think it was a learning experience.

I think you did a fine job. Keep it up and mix up the venues. You will progress.