Should I work out at the same gym that my ex works out at?
Absolutely. Outlast them at the gym. Let them see the body they can no longer "get". Let them see the transformation. My ex-wife looks gorgeous, but then again, so do I. We will not allow our bodies to get out of shape, even though she posts her photos on FACEBOOK and I rarely do. I am not focused on showing them what they are missing, rather, I am showing how much I am enjoying my life and taking care of my body and mind.
Do not let your mistakes from your past deny your destiny for a better life.