Let $V$ be the real vector space of continuous functions $f:[0,1] \rightarrow \mathbb{R}$. Show that defining

$$\|f\|=\int_{0}^{1}|f(x)| d x$$

makes $V$ a normed vector space.

Define $f_{n}(x)=\sin n x$ for positive integers $n$. Is the sequence $\left(f_{n}\right)$ convergent to some element of $V$ ? Is $\left(f_{n}\right)$ a Cauchy sequence in $V$ ? Justify your answers.