# a non-decreasing sequence of functions with bounded L^p-norm is a Cauchy sequence in L^p space

Let $\{f_k\}_{k=1}^\infty$ be a sequence in $L^p(\mathbb{R})$ for $1\leq p<\infty$. Suppose $f_1\leq f_2\leq\cdots$ and $\sup ||f_k||_p<\infty$. Prove that $\{f_k\}_{k=1}^\infty$ converges in $L^p$-norm.

Case 1. $p=1$.

Let $g_n=f_n-f_1$. Then $g_n$ non-negative, increasingly converges to a function $g$ pointwise. By Levi's Theorem, $\int g_n\to \int g$. Let $f=g+f_1$. Then $f_n$ converges pointwise to $f$ and $\int f_n\to \int f$. $$\int|g|=\lim_{n\to\infty}\int|g_n|=\sup\int|g_n|\leq \sup \int |f_n|+\int |f_1|<\infty.$$ By Lebesgue dominated convergence theorem, $\lim_{n\to\infty}\int |g-g_n|d\lambda=0$. Thus $$\lim_{n\to\infty}\int|f_n-f|d\lambda=\lim_{n\to\infty}\int|g_n-g|d\lambda=0.$$

Case 2. $1<p<\infty$.

The above argument cannot be applied in this case. What should I do?

Now we separate out the set of point that is eventually positive and those that never be. For point that is eventually positive then eventually $|f_{n}|$ will be increasing, by monotone convergence, the norm of that limit is the limit of the norm. For the rest, then $|f_{n}|$ must be decreasing, you can use dominated convergence (dominated by $|f_{1}|$). Since the norm of the sequence is increasing and bounded, it does indeed have a finite limit and thus is in $L^{p}$.
Following your construction of $g_n$, for $1<p<\infty$, we use the fact that $L^p$ is reflexive Banach spaces. Given $\|g_n\|_p$ is bounded, $g_n \rightarrow g$ pointwise a.e. imples $g_n\rightharpoonup g$ in $L^p$. And from $0\leq g_n \leq g_2 \leq...$ we have $$\limsup\|g_n\|\leq \|g\|,$$ together with weak convergence we get $g_n \rightarrow g$ strongly in $L^p$. Thus $f_n \rightarrow g+f_1$ strongly in $L^p$.