I think a good thing to do is just try and establish eye contact (if you happen to walk by her or be in her area) and then shoot her a little smile.
If she smiles back, this is an invite to approach. I do this myself when trying to gym approach.
Remember, you will see this person multiple times so there is no need to jump the gun and get the number. You want to slowly built a rapport with her and make her comfortable.
Yes and no. I would assume no. It depends upon the time and what times both you and her go. It depends upon whether you or her are thinking about switching gyms. Gym membership lists have a lot of turnover in a given 1-2 year period. You can never guarantee that you'll see her again.
Take these 2 situations....
1. You assume you'll see her multiple times. You don't jump the gun and get her number. You never see her again. You don't want this.
2. You assume this will be the only time you see her and you suggest a date and get her number. She rejects you or you have 1-2 dates that don't result in sex. You end seeing this person multiple times in the gym for the next 6-12 months or beyond. This sucks and is likely the worse outcome than situation 1 above. I had this happen to me.
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I'm more in favor of gym pickups than against them because one must proceed with caution. If you're going to run game at the gym, you have to be ok with either the possibility of changing gyms if an interaction goes sour or some uncomfortable social situations if you stay and an interaction goes sour.
I think you have to consider how much you like your gym. If you like your gym a lot, don't want to switch gyms, and don't want uncomfortable interactions, then try not to run game there unless you get super strong IOIs like
@cola says.
If you attend some big box chain gym like an LA Fitness or 24 Hour Fitness, you can probably run game there all you want and then start going to a different location if some interaction goes bad.
Another option is to have 2 gym memberships. One primary gym where you don't mess around and then a 2nd rotation membership where you go to hit on women. Boutique studios that are often class based are good for this and even ClassPass can be used to rotate secondary facilities for pickup. I've done this at times.