Wikipedia: "Every convergent sequence (with limit s, say) is a Cauchy sequence, since, given any real number ε > 0, beyond some fixed point, every term of sequence is within distance ε/2 of s, so any two terms of the sequence are within distance ε of each other." http://en.wikipedia.org/wiki/Cauchy_sequence
-
Cauchy sequence in $ L^p(\mu) : \forall \epsilon>0 \space{ } \exists N \in \mathbb{N}$ such that $\forall n,m>N \space{} ||f_n(x)-f_m(x)||_p<\epsilon $
-
Pointwise convergent sequence: $\forall \epsilon>0 \space{ } \exists N \in \mathbb{N}$ such that $\forall n>N \space{} |f_n(x)-f(x)|<\epsilon $
-
Uniformly convergent sequence (on set $E$): $\forall \epsilon>0 \space{ } \exists N \in \mathbb{N}$ such that $\forall x \in E, \forall n>N \space{} |f_n(x)-f(x)|<\epsilon $
However $L^p$ norm $(\int_X|f_n-f_m|^p d{\mu})^{\frac{1}{p}}$ is not the same as the absolute value in 2. and 3. So is the Wikipedia claim true?
Best Answer
Well, the definition of Cauchy sequence can be given in any metric space $(X,d)$ as Wikipedia points out, while the notion of converging sequence requires only a topology on a set to be well-defined (see here). So, if you can understand the sketch of proof given by Wikipedia and write it down rigourously, you'll see that it works for every metric space: you just have to substitute the absolute value of the difference of two real numbers, which is the standard metric on $\mathbb{R}$, with the given distance for an arbitrary metric space. $L^{p}$ does not make any difference, since it is a metric space with the distance induced by $\vert\vert\cdot\vert\vert_{p}$ norm.