Example for converges series in the metric space

Question

Give example for converges series in the metric space:

$$ \quad\quad\quad(\mathrm {R}^n,d_{\infty}),d_\infty=\max\mid x_i-y_i\mid$$

My attempt:

Let $\{X_n\}=x_1,x_2,\dots,x_n\quad\quad\text{points series}\\ \{Y_n\}=y_1,y_2,\dots,y_n$

converges if : $\lim \limits_{n\to \infty}(X_n,Y_n)=(x_0,y_0)\quad\quad$

Let's take

$$\left(\frac 1 n+2, \frac{1}{n^2}+3\right) \xrightarrow[n\to \infty]{}(2,3)$$

I am not sure at all that my attempt is correct, how can I find example?

sorry about my english

Answer 1

First off, your point isn't in $\mathbb{R}^n$. If you use;

$$(x_n)_{n \geq 1} = \left(2+\frac{1}{n}, 3+ \frac{1}{n},...,0\right)$$

Then with the $d_{\infty}$ norm you have;

$$\left|\left(2+\frac{1}{n}, 3+ \frac{1}{n},...,0\right)\right| = 3$$

Therefore; $(x_n) \to 3$.