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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:

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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?

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