The current tenant was already in place when I purchased the building, and their lease is set to expire in a few months.

As the new owner, you’ll want the lease and your landlord insurance policy to take effect as soon as you take title to the property.Temporary Lease Considerations:While short-term leases can work well, leasing to a seller long-term often leads to complications.

You would need to notify them in writing that you will not be expending their lease and they will not be transitioning to a month-to-month lease. 

Can you lease an apt under USDA low income and have someone else live there???

All things being equal, a simpler option would be to terminate the lease with a penalty of one month's rent, agreed to in writing.

Number one, do you have a comp from another land lease property.

@Hai Le - Usually you have to provide written notice / 30 day notice if you decide to not renew their lease or even just asking a tenant to leave.

Our lease break fee is 2 months of rent.

Quote from @Sumi Devi: One of my clients is buying commercial real estate property with tenants that have 5 year lease.

If they don't agree, then you'll have to wait until the current lease expires and then move them to your new lease.Read the existing lease closely.