Let $V$ be the vector space of continuous real-valued functions on $[-1,1]$. Show that the function

$$\|f\|=\int_{-1}^{1}|f(x)| d x$$

defines a norm on $V$.

Let $f_{n}(x)=x^{n}$. Show that $\left(f_{n}\right)$ is a Cauchy sequence in $V$. Is $\left(f_{n}\right)$ convergent? Justify your answer.