A not Banach space

Question

Please, how to choose a sequence to prove that this space is not a Banach space $(\mathcal{C}([0,\frac12],\mathbb{R}),||.||_1)$.

In the exercice they present this sequence of functions $f_n(x)=\frac1n\frac{1}{\sqrt{1-x^2}}$

but this sequence is Cauchy and it converge to 0, so i can't use it to deduce that this space is not Banach!

Answer 1

Defined $(f_n)_{n\in\mathbb N}$ such that:

• $f_n(x)=0$ if $x<\frac14-\frac1{4n}$;
• $f_n(x)=1$ if $x>\frac14+\frac1{4n}$;
• $f_n$ is affine on $\left[\frac14-\frac1{4n},\frac14+\frac1{4n}\right]$.

Then $(f_n)_{n\in\mathbb N}$ is a Cauchy sequence, but it doesn't converge on $\mathcal C\left(\left[0,\frac12\right]\right)$.