For the metric space described below, find the distance between two different open balls of the same radius r if both balls are subsets of a closed ball of radius r.
The metric space was given by the previous problem which said "In set X of all possible sequences $x=(n_1,n_2,...), y=(m_1,m_2,...)$ let $k_0(x,y)$ be the smallest index $k$ such that $n_k \neq m_k$. Then we had to prove a bunch of things which may be useful here
a) $\rho(x,y) = k_0(x,y)^{-1}$ is a metric
b) triangle inequality is strengthened as $\rho(x,z) \leq$ max{$p(x,y), \rho(y,z)$}
c) for any open ball $B_r(x) = \bar{B}_r(x)$
d) $B_r(x)=B_r(y)$ for any $y \in B_r(x)$
e) if two balls have a common point then one of them is a subset of the other ball
f) $x_n$ is a Cauchy sequence iff $\rho(x_n, x_{n+1}) \rightarrow 0$ as $n \rightarrow \infty$
g) X is complete and separable
I've been able to prove most of these so far, but I'm not asking about them, I just think they might be useful for this question. I don't know where to start
If property d) holds, then all three balls in question are the same (also using c)).
Cheers