Using calculus to show that $f_n(x)=x^n$ is not Cauchy in $C^0[0,1]$

Question

As the title says, I need to prove "using calculus" that the sequence of functions $f_n(x)=x^n$ is not Cauchy in $C^0[0,1]$.

The thing that came to my mind is to use the $L_1$ or $L_2$ norm since there are some integral calculations in it and thus it'd be "using calculus." But then I saw it'd prove the opposite in that the given sequence IS Cauchy. At this point I think only using the max(infinity) norm would give me the proof. But would that be "using calculus?"

Answer 1

Here's one possible solution, which I leave you to complete. What is the maximum value of $f_n-f_{2n}$?