Let $x,y \in \mathbb{R}^n$ be two fixed points. Is there an easy proof of the fact that $$A(r):=\frac{ \text{Vol}(B(x,r) \cap B(y,r))}{\text{Vol}(B(x,r))}$$ tends to $1$ when $r \to \infty$. I measure volumes via the standard Lebesgue measure.
I know that there are formulas for the volumes inscribed in spherical caps, but I wonder if there is an easier way to derive the asymptotic result without going through the exact formulas. Since I only care about the asymptotic behaviour when $r \to \infty$, this might be doable in a more elementary way.
An alternative equivalent formulation, which might seem more intuitive is the following: Set
$$B(r):=\frac{ \text{Vol}(B(x,r) \setminus B(y,r))}{\text{Vol}(B(x,r))}$$
Then $B(r)$ tends to zero when $r \to \infty$, as the distance between the points $|x-y|$ becomes negligible compared to the radius $r$.
Let $d =|x-y|$, then for $r \gt d$, $$B(x,r) \setminus B(y,r)\subseteq B(x,r) \setminus B(x,r - d)$$
So $$\text{Vol}(B(x,r) \setminus B(y,r)) \le \text{Vol}(B(x,r) \setminus B(x,r - d)) = C(r^n - (r-d)^n)$$ for some constant $C$. Thus $$B(r) \le \frac {C(r^n - (r-d)^n)}{Cr^n} = O\left(\frac 1r\right)$$ and therefore goes to $0$ as $r \to \infty$.