Consider a Banach space $(X, ||\cdot||)$ and a compact mapping $f: \overline{B}_1(0) \rightarrow X$ such that $||f(x)||\leq 1$ for all $||x|| = 1$, where $\overline{B}_1(0)$ denotes the closed unit ball in $X$. Show that $f$ has a fixed point.
My attempt:
Since $f$ is compact, then $f(B_1(0))$ is relatively compact in $X$. In particular, it is bounded, hence there exists $r>0$ s.t. $f(B_1(0)) \subset \overline{B}_r(0)$. Define $R: = \max\{r, 1\} > 0$. Thus, $f(\overline{B}_1(0)) \subset \overline{B}_R(0)$.
My idea now, was to apply the Schauder's fixed point theorem to an auxiliary compact self-mapping $g$ on $\overline{B}_R(0)$ to establish the existence of a fixed point of $g$. The problem is that I'm not able to find such an auxiliary function which permits to conclude the existence of a fixed point of $f$.
Any suggestions? Thanks in advance!