Originally Posted by

**surjective** I am considering the sequences $\displaystyle (f_{n})$ of continuous functions $\displaystyle f_{n}:[-1,1]\to \mathbb{R}$ defined by:

$\displaystyle

\begin{displaymath}

f_{n}(x) = \left\{

\begin{array}{lr}

1 & : -1\leq x \leq 0\\

1-nx & : 0 \leq x \leq \tfrac{1}{n}\\

0 & : \tfrac{1}{n} \leq x \leq 1

\end{array}

\right.

\end{displaymath}

$

How do I show that $\displaystyle (f_{n})$ is a cauchy-sequence in the normed vector space $\displaystyle (C([-1,1],\mathbb{R}), \Vert \cdot \Vert_{1})$ and how do I examine if the mentioned normed vector-space is a banach space (i know that every finite dimensional normed vector space is a banach space).

$\displaystyle (C([-1,1],\mathbb{R}), \Vert \cdot \Vert_{1})$ is the normed vector space of continuous real-valued functions defined in the closed interval $\displaystyle [-1,1]$.

Thanks.