这是一份liverpool利物浦大学MATH430的成功案例

Given $A, B \subset \mathbb{R}^{n}$, we define the distance between $A$ and $B$, denoted by $d(A, B)$, as
$$
d(A, B)=\inf {|u-v|: u \in A \text { and } v \in B} .
$$
Let $K, F \subset \mathbb{R}^{n}$ be sets such that $K$ is compact, $F$ is closed and
$$
K \cap F=\emptyset .
$$
Prove that $d(K, F)>0$ and there exist $u_{0} \in K$ and $v_{0} \in F$ such that
$$
d(K, F)=\left|u_{0}-v_{0}\right| .
$$

MATH430 COURSE NOTES ：

$$
u=(p \cdot x)^{2}-p^{2} x^{2}
$$
so we can evaluate the integral by either of the substitutions
$$
x^{2} \rightarrow 0, \quad p \cdot x \rightarrow \sqrt{u} \quad \text { or } \quad p \cdot x \rightarrow 0, \quad x^{2} \rightarrow-u / p^{2}
$$
We’ll consider now the latter choice. (The former gives the same result: See the exercise below.) Again, since we need to Taylor expand in $x$ anyway to find the result for a particular numerator, we expand and perform the $\lambda$ integration:
$$
\mathcal{A}{2}=\sum{n=0}^{\infty} \frac{1}{n !}\left(\frac{u}{2 p^{2}}\right)^{n}\left(\frac{1}{8} p^{2}\right)^{n+D / 2-2} \Gamma\left(2-\frac{D}{2}-n\right) \int_{0}^{1} d \beta\left(1-\beta^{2}\right)^{n+D / 2-2}
$$
Performing the change of variables $\beta^{2}=\gamma$ to convert the remaining integral to a Beta function, and using the identities
$$
\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}, \quad \Gamma(z) \Gamma(1-z)=\pi \csc (\pi z)
$$