If a sequence in metric space $(\mathbb{R}^n)$ with euclidean norm is Cauchy, then is it element-wise cauchy?

Question

So let's say I have metric space $(\mathbb{R}^n, d)$ where $n \in \mathbb{N}$ or $n$ is infinity. We also have $$d(x,y) = \sqrt{\sum_{i = 1}^n|x_i - y_i|^2}$$ which is essentially the Euclidean norm.

Now let's introduce a sequence $(x_n)_{n \in \mathbb{N}}$ in $\mathbb{R}^n$ that is Cauchy. Is it also element-wise Cauchy?

Also if $(x_n)$ is convergent, then is it also element-wise convergent?

Thanks in advance!

Answer 1

For any $i \in \{1, \dots, n \}$, we have

$$\lvert x_i - y_i \rvert \le d(x,y).$$ From there, you can easily prove that if $\{x_n\}$ converges to $l=(l_1, \dots, l_n)$, then the $i$-th coordinate of $\{x_n\}$ converges to $l_i$. And a similar things for Cauchy sequences.