$\{x^n\}$ in $C[0,1]$

Question

$\{x^n\}$ in $C[0,1]$ is Cauchy. Does it converge to zero in $c[0,1]$? I know that this sequence converge to zero in the space $L^1[0,1]$.

The pointwise limit is discontinuous, so it cannot converge in the space of continuous functions. — Dave, Dec 27 '19 at 06:23

score 1 · Accepted Answer · answered Dec 27 '19 at 06:22

1

The point-wise limit is $0$ for $x<1$ and $1$ for $x=1$. This limit is discontinuous. Hence the convergence cannot be uniform.

answered Dec 27 '19 at 06:22

Kavi Rama Murthy

1 Answers1