Received A Cease And Desist Letter From Downstair Neighbor Against My Child

Location: NYC
I just received a Cease and Desist letter from downstair neighbor’s attorney accusing my 4 years old son aggressively stomping and running and hammering and playing with balls all day, and kicking her doors and shouting in the hallway and asked us to respond within 7 days saying that we will comply. All of those accusations were untrue. We are a working family, and have been living here for almost 6 years, we are very respectful with neighbors, and had no prior complaints with anyone else. My son goes to school 8am-6pm and sleeps before 9-9:30pm. We also try many outdoor times after school and during the weekends. The downstairs moved in last year and ever since they made our live hell by banging the ceiling when my son is playing at home. We got so stressed because we had to keep on reminding my son to be quiet but he is just a normal 4 years old and he is even not home all the time only 3-4 hours a day at most. The coop management checked our carpets and everything is in compliance. Recently there were some older kids got inside of our building and running downstairs and kicking people doors, the neighbor immediately thought that it was my son. My son never even took the stairs, and he is afraid of heights so he definitely wouldn’t running downstairs. And another resident from the building witnessed everything and he said he would be my witness to make a statement if possible. Regarding the letters we received, I know that she is bullying us and trying to scare us without any proofs. What else i can do now? Should i hire an attorney?