Search Results: ways to approach guys at a gym

I see the same hot guy everyday on my way into the gym, we walk by each other on different sidewalks as we both go in different directions. He blatantly stares at me when I walk by and now I have started looking at him back and he still stares back. I usually look away first super shy. This has been going on for weeks now. Before I started looking back at him we used to pass by much closer and I'd freak out and look at the ground. How can I get him to approach me apart from looking back and smiling? I haven't smiled yet as he's been too far away. My worry is I won't see him for much longer as I will be moving house and won't be going the same way I have been. We are both late 20s he's maybe a year or two younger than me. Is it too highschool like to give him a note because I don't have the balls to say anything? that way if he rejects me I don't need to see him again. Also, I don't want to come off as desperate or easy... Is it a turn off for a woman to approach a guy? Would he be staring if he wasn't interested? How long does a guy take to approach?

If the note is a good idea, any clues on what to say in it?

I kinda feel like if he's not going to approach the note is my only option and what have I got to lose since I won't need to see him again... and if I do nothing I'll always wonder but at the same time I don't want to look desperate.

Well, I met this guy in my french class, when I saw him I totally got nervous, I thought he was really hot so I stared at him many times lol We had like 3 classes together because our groups were separated so once he looked at me and I looked at him back and he turned his face quickly, like "damn, busted" lol.

I thought I would never see him again because they were just two/three classes... but, I FOUND HIM IN THE GYM I go to, I didn't know what to do honestly, I got super nervous like "this is not even possible, why is he even here?". After a while I said "hi", he did the same but it was totally casual.
I'm a stalker so I found his FB account but nope, I won't send him a friend request because we don't know yet. What I found is that he's single and that he's a good guy.

Should I talk to him more? Should I try something? Should I forget him? Should I let him do the approach?

by the way, he's 23, I'm 16. In Mexico the whole age thing is not like many other countries, not a big deal, but, what you think, is it too much?