Let $X$ be uniformly convex Banach space. $f:K\rightarrow K$, such that $\parallel fx-fy\parallel \leq\parallel x-y\parallel\,\,\forall x,y\in K $, with $K$ a nonempty, closed, convex, bounded subset of $X$.

Set $C_{\varepsilon}=\{x:\parallel x-fx\parallel\leq\varepsilon\}$, where $a=\lim\limits_{\varepsilon \rightarrow 0}\inf\limits_{C_{\varepsilon}}\| x\|$.

I want please to prove that the intersection of all sets $C_\varepsilon$ is nonempty (line 11, page 382).[https://www.researchgate.net/publication/38323269_An_elementary_proof_of_the_fixed_point_theorem_of_Browder_and_Kirk].

Thank you.

Let $X$ be uniformly convex Banach space. $f:K\rightarrow K$, such that $\parallel fx-fy\parallel \leq\parallel x-y\parallel\,\,\forall x,y\in K $, with $K$ a nonempty, closed, convex, bounded subset of $X$.

Set $C_{\varepsilon}=\{x:\parallel x-fx\parallel\leq\varepsilon\}$, where $a=\lim\limits_{\varepsilon \rightarrow 0}\inf\limits_{C_{\varepsilon}}\| x\|$.

I want please to prove that the intersection of all sets $C_\varepsilon$ is nonempty; see line 11, page 382 of this paper.

Thank you.