You can only convince people that are open to being convinced. Usually I've seen this take some sort of harsh real-world personal experience that contradicts their existing world-view.

Personal example: I used to strongly defend Obamacare, and would dismiss any arguments against it as trolling. Until I had to sign up with Covered California, deal with the buggy website, wait on hold for hours, then deal with higher deductibles and miscellaneous bureaucracy. Then I realized the detractors had a point and basically switched sides.

Lastly, do you have to persuade someone to take your point of view? Are you able to be friends with them otherwise? If so, continue to be their friend and wait until an event happens that leads them to question their beliefs. Then subtly drop hints...remember, nobody likes to be told they're "wrong".