In Advanced Microeconomic Theory, the author mentions that the Cobb Douglas utility function is neither strongly increasing nor strictly quasiconcave over all of R+. But the condition of strongly increasing in R+ is used to prove that set of Walrasian Equilibrium allocations is a subset of Core. So we can't apply this result to Cobb Douglas utility functions to prove that Walrasian Equilibrium obtained is part of core. Is local non satiation sufficient to say set of Walrasian Equilibrium allocations is a subset of Core as Cobb Douglas satisfies local non satiation?
There are two standard ways to define the core of a (finite) exchange economy. The first version says that there is no coalition, no nonempty set of agents, such that everyone in the coalition can be made better off by just using the total endowment of the coalition. The second version requires that there is no coalition such that someone in the coalition can be made better off without making anyone in the coalition worse off by just using the total endowment of the coalition.
Clearly, if the second version holds, so does the first one. That every equilibrium allocation satisfies the second version can be proven exactly as one proves the first welfare theorem. It requires an assumption like local non-satiation, just like the first welfare theorem requires. That every equilibrium allocation satisfies the first version requires no assumption whatsoever; it follows directly from the definition of equilibrium: If one makes everyone in a coalition better off than what they get in an equilibrium allocation, everyone must consume a more expensive commodity bundle. But this implies that the price of the total consumption of the coalition exceeds the price of the total endowment of the coalition, which is not possible if the total consumption rquals the total endowment.