Is this set bounded? (Hilbert spaces)

Question

If $H$ is a Hilbert space, and we have a function $g: H \rightarrow H$, I have to prove if $g$ is a contraction, then the set $G=\{x \in H : \exists \alpha \in (0,1) / x=\alpha g(x)\}$ is bounded.

Why must be $G$ bounded?

Answer 1

Let $g$ satisfy there is $\rho$: $\|g(x)-g(y) \|\le \rho\|x-y\|$ for all $x,y$. By $$ x = \alpha g(x) = \alpha (g(x)-g(0) + g(0)) $$ we have $$ \|x\| \le 1\cdot \rho \|x-0\| + 1\cdot\|g(0)\| $$ and so $\|x\| \le \frac1{1-\rho}\|g(0)\|$