# show by induction that $\cap_{p}{\text{clco}}\cup_{m\geq p}\frac{1}{m}\sum_{n=1}^{m}{C_n}\subset \cap_{p}\text{clco}\cup_{n\geq p}{C_n}.$

Let $$X$$ be a separable Banach space, and we consider the collection: $$\mathcal{P}_{wkc}(X)=\{C\in 2^X: C \text{ is nonempty convex weakly compact subset of }X\}$$ By $$w$$ we shall indicate the weak topology on $$X$$. We consider the following lemma:

with: $$\text{clco}(A)=\text{cl}\bigg(\left\{\sum _{i=1}^{n}\lambda _{i}x_{i}:n\in \mathbb {N} ,\,x_{i}\in A,\,\sum _{i=1}^{n}\lambda _{i}= 1\right\}\bigg).$$

How to follow the proof of induction in this case?