Proof that two balls intersect

Question

Prove that the two open balls (,0.9) and (,0.9) which are both contained in (0,1) intersect. I have tried to prove they both contained zero but it doesn't look like they do and I am getting confused on how to apply the distance function. We are looking at the metric space (,1,2∞)

Answer 1

Let $v$ be a unit vector orthogonal to $x$. The fact that $B(x,0.9) \subset B(0,1)$ implies that $x+rv \in B(0,1)$ if $0<r<0.9$. Thus $\|x+rv\|^{2}=\|x\|^{2}+r^{2} <1$ if $0<r<0.9$. It follows that $\|x\|^{2} \leq 1-(0.9)^{2}=0.19$ and $\|x\| <\sqrt {0.19}$. Similarly $\|y\| <\sqrt {0.19}$. Using this check that $x \in B(y,0.9)$ so $B(x,0.9)$ and $B(y,0.9)$ intersect.