How to show the convergence in $L^p$ spaces?

Question

Let $f_{n}\in L^{p}(\Bbb{R})\ (1\le p\le\infty)$. For $0<r<1$, if there exists a positive constant $C$ such that $||f_{n+1}-f_{n}||_{p}<Cr^{n}$, how to show $(f_{n})$ converges in $L^{p}(\Bbb{R})$?

Answer 1

$\newcommand{\norm}[1]{\left\Vert{#1}\right\Vert} \newcommand{\nrm}{\norm{\bullet}} \newcommand{\R}{\mathbb R}$

Recall this theorem:

Let $(V,\nrm)$ a normed $\R$-vector space. The following are equivalent:

1. $(V,\nrm)$ is a Banach space
2. For all $\{f_n\}_{n\in\mathbb N}$ such that $\sum_{n\in\mathbb N}\norm{f_n}<\infty$, there exists $g\in V$ such that $$\lim_{n\to\infty}\norm{g-\sum_{k=0}^nf_k}=0$$

Now, in your case $$f_n=f_0+\sum_{k=1}^n(f_{k}-f_{k-1})$$ And $\norm{f_0}_p+\sum_{k=1}^\infty\norm{f_k-f_{k-1}}_p\le\norm{f_0}_p+C\cdot\sum_{k=1}^\infty r^{k-1}=\norm{f_0}_p+C\frac{1}{1-r}<\infty$

Therefore, since the $(L^p(\R),\nrm_p)$ is a Banach space, there exists $g\in L^p(\R)$ such that $$\lim_{n\to\infty}\norm{g-f_0-\sum_{k=1}^n(f_k-f_{k-1})}_p=\lim_{n\to\infty}\norm{g-f_n}_p=0$$ $\square$

Answer 2

We can show that $\{f_n:n\ge1\}$ is a Cauchy sequence. Suppose that $n>m$. We have that \begin{align*} \|f_n-f_m\| &=\|f_n-f_{n-1}+f_{n-1}-f_{n-2}+\ldots+f_{m+1}-f_m\|\\ &\le\|f_n-f_{n-1}\|+\|f_{n-1}-f_{n-2}\|+\ldots+\|f_{m+1}-f_m\|\\ &=\sum_{i=m}^{n-1}\|f_{i+1}-f_i\|\\ &<C\sum_{i=m}^{n-1}r^i. \end{align*} Since $0<r<1$, the series $\sum_{i=0}^\infty r^i$ converges and the term $\sum_{i=m}^{n-1}r^{n}$ can be made arbitrarily small by choosing large $m$. Hence, the sequence $\{f_n:n\ge1\}$ is a Cauchy sequence.