After she makes eye contact for a while and smiles can I walk up to her and introduce myself?

Anonymous
This happens a lot and I know I should approach a girl after we make eye contact and she smiles but I never know what to say. Like for example I was at the library on campus using a computer last week and this girl sat across from me and I kept catching her looking at me. She looked away pretty fast when I caught her a few times so I think she was checking me out. I wanted to start talking to her but had no idea how to break the ice. So when something like that happens can I just go up to her and say something like:

"Hi I'm Joe Bob , what's your name?"

And then take it from there. Or would that be really random?
After she makes eye contact for a while and smiles can I walk up to her and introduce myself?
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