Tenant "says" they gave last landlord last month's rent

[Tony Murphy]

So I just bought my first rental property, a duplex, from an elderly owner and was given the security deposits for both of the current tenants at closing. When I went to reintroduce myself to them, one of the tenants asked about her security deposit and the last months rent. According to the previous owner, he only has security deposits. There was no last month's rent given and the tenant says she can't find her paperwork showing that she actually paid it. She's been there 7yrs and neither the owner or she kept very good written records. Will I be responsible for the money if she decides to move out in a few years and is able to find the paperwork then, especially if God forbid, the previous owner has died? Would she need to go after his estate? I appreciate the help!