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


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; $(x_n) \to 3$.

  • $\begingroup$ why $(x_n)\to 3$? $\endgroup$ – Error 404 Mar 16 '16 at 11:07
  • $\begingroup$ The max difference between any of the coordinates is $3+ \frac{1}{n} - 0$ which goes to $3$ as $n \to \infty$. $\endgroup$ – Faraad Armwood Mar 16 '16 at 13:57

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.