Would I be stupid for staying?

I need honest answers 😑 my boyfriends ex told me that he had been speaking to her 'begging' to meet up and stay over at his etc. I had text her asking when the last time she spoke to him because I saw that he still had her on Facebook. I asked her for proof but she said it was on snap chat so it got deleted. I sent him the screen shots of what she said and blocked him straight away and didn't unblock him for a few days. (we text on Facebook). I have instagram and he doesn't, so he made an account and kept texting saying she's lying. He said he left her just before getting back with me and she didnt like it so she saw a perfect opportunity to f*ck him over. He kept say how I shouldn't let some 'lying h*e' mess up our relationship. I ignored these messages and still didn't text him until yesterday asking him to tell me the truth etc and he still denies it. Not only do I love him more than anything but we have an 8 week old baby and I really don't want my little family to break up if the girl is actually lying. I just don't know what to do. Would I be stupid for staying? Would you break up with your husband/boyfriend on some girls say so without any proof?? Especially if you have kids??