= Completeness of Lp spaces
{title2=$1\leq p<\infty\ \Longrightarrow\ L^p\text{ is Banach}$}
From an <Lp norm> Cauchy sequence choose a subsequence with successive difference norms at most $2^{-j}$. <Minkowski inequality> and <monotone convergence theorem> make the sum of the absolute differences an $L^p$ function, hence finite almost everywhere. The telescoping series defines an $L^p$ limit; the geometric bound on its tail proves norm convergence of the subsequence, and the original Cauchy property proves convergence of the whole sequence. No sigma-finiteness assumption is needed.
Back to article page
