This is part $2$ of a $4$ part question.
Part $1$ (and links to other parts)
The body of the problem is below:
If $G \times X \to X$ is a group action then the kernel of the action, $K$, is the intersection of all the stabilizers $$K = \bigcap_{x \in X} \text{stab}_{G}(x)$$
Let $\Psi$ be the collection $$\Psi = \{ \text{stab}_{G}(x) \}_{x \in X}$$ Then, by definition of the intersection of a collection of sets, we have: $$\bigcap_{\text{stab}_{G}(x) \in \Psi} \text{stab}_{G}(x) = \{ g \in G \mid g \cdot x = x, \forall x \in X \}$$ Which is exactly the definition of the Kernel of the action.
What do you guys think? Does this seem to be alright. The proof seems quite simple. All it is, is an application of the definition of the intersection of a collection of sets, so it seems like there should be more.