Prove my babys father is untrustworthy??

Document everything, get character statements from others, stand firm for what you know is right for your child. Also, don't start arguments, etc with him over his behavior just mentally note what you see/hear with your own eyes/ears and then document it in writing or to your Legal counsel. Hope this helps.

You need to find something that the father is doing that is detrimental to the well-being of your baby. Just the fact that he doesn't exclusively date one woman isn't grounds for the denial of a father's rights.