How to prove that $f_n(x):=x^n$ is not a Cauchy sequence in $C[0,1]$ under the norm $\|f\|= \sup|f(x)|$? 
How to prove that $f_n(x)=x^n$ is not a Cauchy sequence in $C[0,1]$ under the norm $\|f\|= \sup_{x\in [a,b]}|f(x)|$, by showing that it does not satisfy the definition of a Cauchy sequence?


Fix $N\in\mathbb{N}$. Note that:
$$ \| f_N-f_{2N}\|=\sup_{0\le x\le 1} |x|^N|1-x^N|\ge (2^{-1/N})^N|1-(2^{-1/N})^N|=2^{-1}(1-2^{-1})=1/4$$
Since $N$ was arbitrary, this shows that the definition fails with $\varepsilon=1/4$. 
