Inheriting a Tenant

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I just got my first property under contract, it's the first property I've ever purchased including primary residence. It's a 2 unit building and I'm moving into one and renting the other. 

The sellers agent says the current tenant isn't in a lease and pays monthly rent between $1100 to $1200.

Im approaching this tenant for the first time. I want to raise rents and ask him/her to sign a lease. The market indicates I can get about $1300 for this unit.  

Is there a way to do this without ruffling feathers? Any advice is appreciated.

[Bill P.]

@Darren Samson assuming she's been a good tenant, I'd draw up a 12 month lease for 1300 but give her a 6 month discount of 100 so everybody has gotten something fair