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

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数学分析 Analysis MATH41220-WE01/MATH1051-WE01/MATH3011-WE01
问题 1.

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
$$
L^{\prime}=\lim {n \rightarrow \infty} d\left(p{n}^{\prime}, q_{n}^{\prime}\right) .
$$


证明 .

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
$$
\left|d\left(p_{n}, q_{n}\right)-d\left(p_{n}^{\prime}, q_{n}^{\prime}\right)\right| \leq d\left(p_{n}, p_{n}^{\prime}\right)+d\left(q_{n}, q_{n}^{\prime}\right) .
$$

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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
$$
d_{p}(x, y)=|x-y|_{p} .
$$








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