Tenant Didn't Pay- Wierd Senario

[Anna Buffkin]

So I closed on a duplex Friday.  I had a sick child so my husband closed in my absence, so this is all being relayed second hand, but here goes.  

Before closing, I made sure prorated rent and security deposits were included on the HUD; however, the seller shows up at closing and said they just found out that tenant B had not paid their rent this month per the property manager. The closing attorney pretty much told them that the time to dispute the closing statement was passed unless they want to move the closing. If they wanted to close today, they should accept the closing HUD statement and get their check. They came from out of town and went ahead and closed.

Here is my issue.  How do I deal with the new tenant?  The seller basically paid my portion of their rent owed and it's not my problem that they basically stiffed the seller a month's rent.  I feel like trying to collect on it would be like double dipping unless I want to track down the seller if they paid and their is no upside for me in trying to collect in that case.  I don't think I can start an eviction as technically I was paid their rent by someone (the seller).  I am leaning toward just starting fresh on Dec 1 and keeping them on a short lease (i.e. start the eviction as soon as I legally can on day 5 rather than waiting until my customary day 10.)  When I toured the unit before purchase, the tenant keep the place very clean and no issue beyond nonpayment appears to be going on as far a lease violations.    

Thoughts?