Question about proof of Browder, Minty Theorem

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Could someone please assist with the following question:

Refer to the following post.

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Since $e_1,\ldots,e_n$ are fixed, the vector $c^n$ is simply the vector of the components of $u_n$ in the basis $\{e_1,\ldots,e_n\}$. Thus, considering the vector $u_n$ or $c^n$ is completely equivalent.