Is the sequence of functions, $f_n(x) = x^n$ Cauchy in $C([0,1])$? Is it Cauchy in $L^2([0,1])$?
I know that it is not Cauchy in the space of continuous functions. I know that the L-space is complete and suspect that the sequence will be Cauchy in that space then.
My idea to prove it is by contradiction. I have not been provided with a particular norm to work with in the continuous function space.
The norm in the L-space is
$\left\lVert f\right\rVert = (\int_a^b x(t)^2 dt)^{1/2}$
I'm wondering what the distinguishing factor between the sequence being in either space will be.
It must be that the sequence does not converge to a continuous function in the first case. I'm not sure what this function will be.
Best Answer
If it were Cauchy in $C[0,1]$ then it would converge. This is convergence in uniform sense and the limit function would be continuous. The limit function would have to agree with the pointwise limit function.
However, it converges pointwise to a discontinuous function which is 0 except at 1 where it is 1.
For $L^2$ the question is already answered.