Let $V$ be the vector space of all sequences $\left(x_{1}, x_{2}, \ldots\right)$ of real numbers such that $x_{i}$ converges to zero. Show that the function

$$\left|\left(x_{1}, x_{2}, \ldots\right)\right|=\max _{i \geqslant 1}\left|x_{i}\right|$$

defines a norm on $V$.

Is the sequence

$$(1,0,0,0, \ldots),(0,1,0,0, \ldots), \ldots$$

convergent in $V ?$ Justify your answer.