# $L^p$ convergence proof check

I don't have much experience with measure theory, so I want to make sure that I'm not making any bad mistakes. I also want to be sure that the theorem is true so I can use it.

Theorem: Let $\{u_i\}$ be a Cauchy sequence in $L^p(U)$. Then for all $\epsilon>0$, there exists an $N$ such that $|u_i(y)-u_j(y)|<\epsilon$ almost everywhere in $U$ for all $i,j>N$.

Proof: Suppose not. Then there exists an $\epsilon$ such that for all $N$ there is an $i,j>N$ such that $|u_i(y)-u_j(y)|\ge \epsilon$ for $y\in S \subset U$ with $\mu(S)>0$.

Since $\{u_i\}$ are Cauchy in $L^p$ we have $\int_U |u_i(y)-u_j(y)|^p < |S|\epsilon^p$ for $i,j>M$. But $|S|\epsilon^p = \int_S |u_i(y)-u_j(y)|^p \le \int_U |u_i(y)-u_j(y)|^p$, a contradiction.

Thank you in advance.

-

Your theorem is wrong (you proved convergence even in $L^\infty(U)$). The problem in your proof is, that your set $S$ (and, hence, it's measure) depends on $i,j$. Therefore, you can't choose $M$ as desired.