Formal:
* Persuade
* Induce
* Impress
* Satisfy
* Assure
* Demonstrate
* Prove
Informal:
* Sell
* Talk into
* Win over
* Get someone on board
* Make someone see it
* Change someone's mind
More specific:
* Coerce: to force someone to do something by using threats or pressure
* Entice: to persuade someone to do something by offering them something desirable
* Cajole: to persuade someone to do something by using flattery or coaxing
The best synonym will depend on the specific situation and the desired tone.