We consider the space [tex]C^0 ([-1,1])[/tex] of continuous functions from [tex][-1,1][/tex] to [tex]\mathbb{R}[/tex] supplied with the following norm:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]||f||_1 = \int_{-1}^{1} |f(x)| dx [/tex]

a. Show that [tex]||.||_1[/tex] defines indeed a norm.

b. Show that the sequence of functions [tex](f_n)[/tex], where

[tex]

\begin{align*}

f_n(x) &= -1, \quad & -1 \leq{x} \leq{\frac{-1}{n}} \\

\ &= nx, \quad & \frac{-1}{n} \leq{x} \leq{\frac{1}{n}} \\

\ &= 1, \quad & \frac{1}{n} \leq{x} \leq{1}

\end{align*}[/tex]

is a Cauchy-sequence with respect to the given norm.

c. Show that [tex]C^0 ([-1,1])[/tex] is not complete with respect to the given norm.

I figured out a. myself, by showing this norm satisfies the properties of a norm, but I can't find out how to tackle b. and c.

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# Homework Help: Cauchy sequence

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