Why is every action considered to be in the support while calculating mixed strategy equilibrium? While evaluating the mixed strategy equilibrium, in most of the cases, we consider that all actions of a player are in the support of the equilibrium.Is it always so? 
I was going through a problem (36.1, Chapter 3 in Ariel Rubinstein’s A Course in Game Theory) where the mixed strategy equilibrium was calculated by considering all the possible actions in the support of the probability distribution. I was able to reach to this solution using the basic theory, but I cannot prove that every action will be in the support.
Here I am borrowing the definition of support from Ariel Rubinstein’s A Course in Game Theory.
It is not, only in intro problems.

You have already seen counterexamples: The prisoner's dilemma has an equilibrium, yet it does not have full support. (But it is a pure strategy equilibrium! So? All pure strategies are just very particular mixed strategies.)

Mixed strategies with full support are called completely mixed strategies.

It is a good exercise to show that strictly dominated pure strategies are never in the support of mixed equilibrium strategies. Using this you yourself can construct a simple example where the only equilibrium is a mixed but not completely mixed strategy profile.