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?


For a fixed $n \ge 1$ and for $m \ge n$,
$$
             \|x^n-x^m\|=\max_{x\in[0,1]}(x^n-x^m) \ge (x^n-x^m)|_{x=1/2^{1/n}}
         = \frac{1}{2}-\left(\frac{1}{2}\right)^{m/n},
$$
Hence,
$$
         \|x^n-x^m\| \ge \frac{1}{4} \mbox{ whenever } \left(\frac{1}{2}\right)^{m/n} \le \frac{1}{4},
$$
which occurs whenever $m/n > 2$, or $m > 2n$. So $\{ x^n \}_{n=1}^{\infty}$ cannot be a Cauchy sequence in $C[0,1]$.
