Cauchy Convergence in Normed Vector Spaces

[Adblax]

Homework Statement

Fix a<b in R, and consider the two norms Norm(f)₁:=Integral_a^b( Modulus(f)) and Norm(f)_Infinity:= sup{Mod(f(x)): a <= x <= b} on the vector space C[a,b]. This question shows that they are not equivalent.
a. Show that there is K in R such that for all f in C[a,b],
Norm(f)₁ <= K*Norm(f)_Infinity
b. Show that there is no K in R such that for all f in C[a,b],
Norm(f)_infinity <= K*Norm(f)₁
c. Consider the sequence of functions (f_n) in C[-1,1] given by f_n(x):=x^1/(2n-1)
Is (f_n) a cauchy sequence in (C[-1,1], Norm(f)₁)? Why?
Is (f_n) a cauchy sequence in (C[-1,1], Norm(f)_Infinity)? Why?

Homework Equations

Cauchy Convergence Definition:
For all E>0, There exists M(E)>0, in the natural numbers, such that for all m>=n>=M(E), Norm(f_m-f_n)<E

The Attempt at a Solution

Found the first two parts relatively easy I think, for a taking k to be (b-a) and for b, considering the sequence of functions
f_n(x):= { n²-nx for 0<=x<=1/n
0 otherwise
However, for the proof of cauchy sequences I am at a loss, as we cannot use normal convergence in the normed vector space as it converged to something that isn't continuous. So I am trying to use the definition of cauchy convergence, for the first part I have tried splitting up the integral into [0,1] and [-1,0] but have no idea for the second part.

 

[micromass]

For the 1-norm this is a Cauchy sequence.
You need to prove that $$\|f_p-f_q\|_1\rightarrow 0$$as $$p,q\rightarrow +\infty$$.
So, you'll need to calculate

$$\|f_p-f_q\|_1=\int_{-1}^1{|x^{1/(2p-1)}-x^{1/(2q-1)}|}$$

and then prove that this expression goes to 0 as $$p,q\rightarrow +\infty$$.

 

[Adblax]

I somewhat understand the first part yes, however I really cannot see what to do for the second part, as I know it doesn't have cauchy convergence, however, intuitively, the supremum tends to 0 eventually, as given an x, they will both tend to 1 as p,q tends to infinity, and thus the supremum tends to 0 (assuming x isn't 0)

 

[micromass]

C([-1,1]) with the sup-norm is complete, thus all Cauchy convergent sequences converges. But if a sequences converges uniformly, then it converges pointswize. So you need to show that the pointswise limit of your sequence is not continuous.