social network graph problem Here is the problem:

There are connected graph with nodes representing a number of people. Each node/person has an opinion on a topic e.g. trump vs clinton, paper books vs kindle, etc

The goal is make every node in a graph share the same opinion, by selecting a specific subset of nodes, in a particular sequence.

If majority of a person A's friends support trump, but person A supports clinton. if person A is selected, his/her opinion will change to trump.

If person's friends opinions are equally divided, then you can decided the selected person's opinion.

I am running out of ideas as how to proof this is achievable. Maybe some of you can give me some pointers.
This is known as majority dynamics. Usually the assumption is that all nodes adopt the majority opinion simultaneously, and this is known as the synchronous model. For an arbitrary tie-breaking rule, this converges either to a fixed point or to a cycle of length 2; see for example pages 5-6 of Ginosar and Holzman's The majority action on in nite graphs: strings and puppets. If you break ties in a biased way, then the dynamic probably always converges.

What you describe is the asynchronous model, in which the majority rule is applied in sequence rather than in parallel. In that case the process always converges. See for example Tamuz and Tesler, though their methods are probably overkill for you, since in your case you get to choose the sequence, whereas in their case the sequence is chosen at random.