Question about Normed vector space. Here is the definition of a normed vector space my book uses:

And here is a remark I do not understand:

I do not understand that a sequence can converge to a vector in one norm, and not the other. For instance: Lets say $s_n$ converges to $u$ with the $\|\|_1$-norm. From definition 4.5.2 (i) we must have that $s_n$ becomes closer and closer to $u$. Why is it that it could fail in the other norm, when it can become as close as we want in the first norm?Are there any simple examples of this phenomenon?
PS:I know that they say we will see examples of this later in the book, but what comes later is too hard for me to udnerstand now.

You can get in trouble if the convergence is not absolute, i.e. $$\sum_n \|x_n\| = +\infty$$ but 
$$\lim_{N\to\infty} \sum_{n\le N} x_n $$
exists.
