Inherited tenant issues

[Meghan Sullivan]

Hello everyone!  So I've already got a post going about my current tenant issues, but I figured that I'd start a new one since some things have changed.  So I recently purchased a double unit with tenants occupying it month-to-month without any form of rental agreement. Upon closing, I met with the tenant and went over a rental agreement line by line, which she signed without argument.

The first month came and went on February 1st and I have yet to receive a full month's rent from her.  She informed me on February 4th that she won't have the rent until the middle of the month. She has been informed of the late fee, after having been given a five day grace period.

Today, February 14th, she informed me that she only has a partial payment for this month's rent and that she is giving me 30 days notice as of March 1st.  She claims that I will be given the rent that I am owed. At any case, this works for me, because I don't want to have to go through a long eviction process if I don't have to.  But I'm wondering if I should still serve the pay or quit in case for some reason she decides that she wants to stay after all.  I don't want to piss her off if she is in fact going to pay me the rent owed, but I also want to make sure that I'm covered if, for some reason, she decides to stay.  I no longer want her as a tenant. It's also good to note that there was no transferred security deposit with the purchase and the agreement states that payments for that don't start until March, so even better reason not to piss her off. 

It's great if she pays the rent owed, but if she doesn't then that is going to be two months rent missed. If she fails to pay rent, is it worth taking her to small claims court in NY? Her rent is $595/month, as well as a small late fee.

Thanks for the guidance!