Is it $f_n(x) = x^n $ cauchy sequence if:
- $d(f,g) = \int_0^1|f(x)-g(x)|dx$
- $d(f,g) = \mbox{sup}_{x \in [0,1]}|f(x)-g(x)|$
What, if $f_n(x) = x^n - x^{2n}$ ?
In first case we have $f_n(x) = x^n $
If we take $ N = \frac{2- \varepsilon}{\varepsilon}$ then for any $m,n>N$ we have that $d(f_n,f_m) < \varepsilon$ so $f_n$ is Cauchy sequence.
In this case I don't have idea. We have that $d(f_n,f_m) = \mbox{sup}_{x \in [0,1]}|x^n - x^m|$ and what next?
In second case we have $f_n(x) = x^n - x^{2n} $
We can take the same $N$ as before and we receive the same result, so it is Cauchy sequence.
I have the same problem as before.
I will grateful if you tell me, if the point 1. is good in these two cases and please help me with point 2. Thanks in advance.
Hint: $C([0,1])$ with the $\sup$ norm is a Banach space, so if $x_n$ is a Cauchy sequence, then it is also a convergent sequence in this norm (i.e. this sequence converges uniformly to some function).
Similarly for the second sequence, if it is Cauchy, then it converges uniformly to a function.
In both cases, identify what the (pointwise) limit of the sequence should be, then show that the sequence cannot converge uniformly. For the second sequence, it may be useful to find the maximum of $x^n - x^{2n}$ on $[0,1]$ (take the derivative, set $=0$, etc.)