Let $\{f_n\}_{n\in\mathbb{N}}$ a Cauchy sequence in $C[0,1]$. then $\{f_n(x_0)\}_{n\in\mathbb{N}}$ is Cauchy

Question

Let $\{f_n\}_{n\in\mathbb{N}}$ a Cauchy sequence in $C[0,1]$. Prove for each $x_0\in[0,1]$ the sequence $\{f_n(x_0)\}_{n\in\mathbb{N}}$ is Cauchy in $\mathbb{R}$

My work

Let $\epsilon >0$
As $\{f_n\}_{n\in\mathbb{N}}$ then exists $N\in\mathbb{N}$ sucht that if $n,m\geq N$ then $d(f_n,f_m)<\epsilon$

Here I'm stuck. Can someone help me?

Answer 1

Hint:

How is the distance $d(f, g)$ in the vector space $\mathcal C[0,1]$ defined?