If the self is neither waking entity nor the dream entity then it proves the fact that, whatever is seen, known, believed and experienced on the base of the waking entity as self is bound to be falsehood

Deeper self-search reveals the fact that, the self is neither the  the waking entity nor the self is dream entity.  If the self is neither  waking entity nor the dream entity  then  it proves the fact that, whatever is seen, known, believed and experienced on the base of the waking entity as self  is bound to be falsehood.  

Thus arguing on the standpoint of the ego or waking entity, which is the false self, is erroneous.  Thus the whole argument has to be based on the formless soul,which is the innermost self.   Thus it is necessary to realize the fact that, the “self” is not the form but the “self” is formless in order to realize the Advaitic or non-dualistic truth.  

Ignorance of the innermost self makes one remain in the realm of physicality. Formless Consciousness or Ataman is the innermost self. Consciousness is the formless substance and the witness of the duality (form, time and space or universe). 

 In ignorance one is bound to experience the duality (universe or waking) as reality.   The dream becomes unreal when waking takes place. Similarly, the waking becomes unreal, when the non-dual wisdom dawns in the midst of duality. 

Nondual wisdom dawns when the waking entity realizes it is not the self but the self is the formless substance and the witness of the three states.  The formless substance and witness of the three states are one in essence and that essence is consciousness. Thus there is no second thing exists other than consciousness, which is the innermost self.  Thus consciousness is the ultimate truth or Brahman.  Realizing the consciousness or Ataman as the innermost self is self-realization. Thus knowledge of the true self is self-knowledge or Brahma Gnana or Atma Gnana.

When wisdom dawns in the midst of duality, one becomes aware of the fact that, the whole experience of diversity (universe or waking or illusion) is created out of consciousness, and sustained by consciousness, and finally dissolves as consciousness. Thus, the true nature of the true self (Ataman, the innermost self) is formless and nondual.