Fixed Point in Uniformly Convex Banach Space and Bounded Subset

Question

The problem is "Let $K$ be a bounded, closed, convex subset of a uniformly convex Banach space $X$ and $F : K \rightarrow X$ be a nonexpansive map with $\inf{||x − F(x)|| : x \in K} =0$ Show that F has a fixed point in K." Because infimum is zero,then there exist sequence $(x_n) \subset K$ such that $||x_n − F(x_n)|| \rightarrow 0$ when $n \rightarrow \infty$. I want to show $(x_n)$ is cauchy sequence, but in the progress I get stuck. I dont know how to use uniformly convex.Help me, please!.