Behavior towards my ex girlfriend who cheated and lied, did I act good?

So I'm 19 year old guy and I was in a relationship with a cute girl, she's 17. After a month she dumps me over text. She was even planning for us to go to the gym together.

2 weeks later I found out out that she has cheated on me and got in a relationship with that guy (17).

I gave her a call so she would know that I know what she has done (in calm voice). Told her that she didn't have to lie and that we could just talk and it would be fine.

3 months later she is still in relationship with that guy.

I started working out in the gym on regular basis. She goes to the same gym. I know that, we were supposed to go there together. One day, while going to the changing rooms I notice her working out. She immediately notices me, looks at me 2-3 times and moves her look away. I did look at her too, but there was no communication.

A week later, after the lifting session, I'm going with my personal trainer to do some crunches and stretching. It's her, 5 m away (like 15 feet). I immediately striked a funny conversation with my trainer. Didn't give her even a look. Passed by her a few times, didn't even say hello to her. Feeling so proud of myself, not giving a **** about someone who betrayed my trust.

I didn't post sad or depressing stuff on instagram, just me being happy and enjoying life.

Did I act good? How should I behave further on?

Also: I lost 8 Kg (17,6 pounds) since we broke up and I do look better. She must have noticed that (all my college friends noticed the positive change in my appearance though).
I was really good with her mom, and if I see her mom somewhere in the town, I guess I should be nice to her, and that I shouldn't mention my ex.

You did fine, you were only dating a month so very little time invested.

You are young you will find new people to date

I think you're behaving correctly, trying to move on as soon as you can..

You were fine. Ignoring her is probably best since she betrayed you. And not publicly posting about your feelings is wise. Best to be happy and move forward, or at least look like you are to salvage some self-esteem and pride.

I agree with the above posters. Also: I agree that you should be nice to her mom if you see her around.