I was proving this statement and I have found a problem. I have taken the retraction $r\colon X\to A$ defined as $r(x)=\frac{2x}{\|x\|}$ and the homotopy:\begin{equation}
H\colon X\times[0,1]\to X,\quad H(x,s)=(1-s)x+sr(x).
\end{equation}
The thing is that I can't prove that $\|H(x,s)\|>1$, in order to know that the homotopy is well defined. Any ideas?