In my text book, it is stated that for a sequence of functions ${x^n}$ in a metric space $C^0([0,1])$, it is Cauchy in $||-||_1$ norm but not in $||-||_\infty$ norm because that would lead to a contradiction.
I am wondering what this contradiction is.
I tried expanding the norm:
$||x^m - x^n||_1 = \int_0^1|x^m-x^n| dx =_{| m<n} \int_0^1(x^m-x^n) dx = \frac{1}{m+1} - \frac{1}{n+1}$, which goes to zero (which is in C) for large m and n, it is thus Cauchy.
but,
$||x^m - x^n||_\infty = lim_{p->\infty} (\int_0^1|x^m-x^n|^p dx)^{1/p} =_{| m<n} lim_{p->\infty} (\int_0^1(x^m-x^n)^p dx)^{1/p}$
I am not really sure how I could get this to show a contradiction, or if it's even the right way to go...