# 数学分析 Analysis MATH41220-WE01/MATH1051-WE01/MATH3011-WE01

Thus $\left(d\left(p_{n}, q_{n}\right)\right)$ is a Cauchy sequence in $\mathbb{R}$, and because $\mathrm{R}$ is complete,
$$L=\lim {n \rightarrow \infty} d\left(p{n}, q_{n}\right)$$
exists. Let $\left(p_{n}^{\prime}\right)$ and $\left(q_{n}^{\prime}\right)$ be sequences that are co-Cauchy with $\left(p_{n}\right)$ and $\left(q_{n}\right)$, and let
$$L^{\prime}=\lim {n \rightarrow \infty} d\left(p{n}^{\prime}, q_{n}^{\prime}\right) .$$

Then
$$\left|L-L^{\prime}\right| \leq\left|L-d\left(p_{n}, q_{n}\right)\right|+\left|d\left(p_{n}, q_{n}\right)-d\left(p_{n}^{\prime}, q_{n}^{\prime}\right)\right|+\left|d\left(p_{n}^{\prime}, q_{n}^{\prime}\right)-L^{\prime}\right| .$$
As $n \rightarrow \infty$, the first and third terms tend to 0 . the middle term is
$$\left|d\left(p_{n}, q_{n}\right)-d\left(p_{n}^{\prime}, q_{n}^{\prime}\right)\right| \leq d\left(p_{n}, p_{n}^{\prime}\right)+d\left(q_{n}, q_{n}^{\prime}\right) .$$

## MATH41220-WE01/MATH1051-WE01/MATH3011-WE01 COURSE NOTES ：

$$|n|{p}=\frac{1}{p^{k}}$$ where $p^{k}$ is the largest power of $p$ that divides $n$. (The norm of 0 is by definition 0 .) The more factors of $p$, the smaller the $p$-norm. Similarly, if $x=a / b$ is a fraction, we factor $x$ as $$x=p^{k} \cdot \frac{r}{s}$$ where $p$ divides neither $r$ nor $s$, and we set $$|x|{p}=\frac{1}{p^{k}} .$$
The $p$-adic metric on $Q$ is
$$d_{p}(x, y)=|x-y|_{p} .$$