Non-constant Cauchy sequence

Question

I need to find an example of non-constant Cauchy sequence in $\mathbb E^2$. The metric in question is $\rho_2$, so Cauchy sequence would be sequence for which following is true: $\sqrt {(x_m - y_m)^2 + (x_n - y_n)^2}$ for $min(m,n) → 0$. My thought is something like: $(1, \frac{1}{2}), (1, \frac{3}{4}), (1, \frac{5}{6}), \dots$. Are my thought correct and if so, how do I write this sequence formally?

Answer 1

Yes your thoughts are correct...

You can write your sequence $(\mathbf{v}_k)_{k \in \mathbb{N}}=\{(1,1/k): k \in \mathbb{N}\}$.

To show that it is Cauchy:

$1/k,k \in \mathbb{N}$ is a converging sequence in $\mathbb{R}$ and thus Cauchy. So for every $\epsilon > 0$ there are some $N$ such that $|1/l-1/(l+m)| < \epsilon$ for every $l \geq N$ and $m \in \mathbb{N}$. In particular $\rho_2(\mathbf{v}_l,\mathbf{v}_{l+m})< \epsilon$ for every $l \geq N$ and $m \in \mathbb{N}$.

Note that for any pair of Cauchy sequences $(x_k)_{k \in \mathbb{N}},(y_k)_{k \in \mathbb{N}} \subset \mathbb{R}$ (or equivalently any pair of converging sequences) the sequence $((x_k,y_k))_{k \in \mathbb{N}} \subset \mathbb{E}^2$ is also a Cauchy sequence.