existence of a unique vector in a Hilbert space

Question

Let $H$ be a Hilbert space, $A:H\rightarrow H$ a continuous, linear map. $x\in H$ and $ 0<r\in\mathbb{R}$, such that $r>dist(x,A(H))$. Show that there exists exactly one vector $y_0\in H$, such that:

1. $\Vert x-A(y_0)\Vert\le r$
2. $\Vert y_0\Vert=\inf\left\{\Vert y\Vert: \ y\in H, \ \Vert x-A(y)\Vert\le r\right\}$

Can we substitute $\inf$ by the $\min$?

I tried to use some strong tools like Hahn-Banach or Banach-Steinhaus but I really can't see how to apply them? Or maybe this problem goes in the other way?

Any hint?