Cauchy sequence: A sequence of real numbers is a Cauchy sequence if $\forall \epsilon >0, \exists N$ s.t $\forall n,m > N, |x_n - x_m| < \epsilon$
Now, for $n \in \mathbb{N^*}$, $f_n : [0,1] \rightarrow \mathbb{R}$ is defined by $f_n(t) = t^{\dfrac{3}{n}}$
Now I am asked the following:is $\{f_n\}_{n \in \mathbb{Z}_{>0}}$ a Cauchy sequence in the vector space of continuous functions $C( [0,1 ], \mathbb{R})$ with the norm $\|x\|_p$
with $p = 1$
with $p = \infty $
My attempt:
I have found $\|f_n- f_m\|_1 = \dfrac{n}{3+n} - \dfrac{m}{3+m}$ and $\|f_n-f_m\|_{\infty} = \Big(\dfrac{n}{m}\Big)^{m/(m-n)}- \Big(\dfrac{n}{m}\Big)^{n/(m-n)}$ on a previous question.
But I have no idea how to show (or refute) that they are Cauchy sequences.