# Convergent sequence such that its components doesn't converge to the components of the limit

I would like to find

• a vector space $$E$$
• a norm on $$E$$
• a sequence $$(u_n)$$ which converges for this norm but such that its components doesn't converge to the component of the limit.

First $$E$$ should be infinite dimensional.

Second I am tying to look at $$E=\mathbb{R}[X]$$.

Take $$u_n=(\underbrace{0,\ldots,0}_{\text{n zeros}},1/n,1/(n+1),\ldots,1/(n+p),\ldots)$$.

or

$$u_n=(1/1,1/2,\ldots,1/n,0,\ldots,0,\ldots)$$.

We have $$||u_n||_1=\sum_{k=1}^n1/k$$

• The question does not make much sense. Please specify the space of sequences you are considering and specify the norm you want to use. – Kavi Rama Murthy Feb 11 at 10:14
• This can be an answer, if I understand the question properly. – Giuseppe Negro Feb 11 at 10:18