**Case 1: $p \in [1, \infty)$.** Let $(f_k)_k$ be a Cauchy sequence in $L^p$. Choose a subsequence $(f_{n_k})_k$ with $\|f_{n_{k+1}} - f_{n_k}\|_p < 2^{-k}$ for all $k$. Define $g_k := |f_{n_{k+1}} - f_{n_k}|$ and $G_N := \sum_{k=1}^N g_k$. By the triangle inequality, $\|G_N\|_p \le \sum_{k=1}^N \|g_k\|_p < 1$. Since $G_N \uparrow G := \sum_{k=1}^\infty g_k$ pointwise with $G_N \ge 0$, the [Monotone Convergence Theorem](/theorems/509) gives $\|G\|_p \le 1 < \infty$. In particular $G < \infty$ a.e., so the telescoping series $f_{n_1} + \sum_{k=1}^\infty (f_{n_{k+1}} - f_{n_k})$ converges absolutely a.e. Define $f$ to be its pointwise [limit](/page/Limit) where it converges, and $0$ elsewhere. Since $|f_{n_{K+1}} - f_{n_1}| = |\sum_{k=1}^K (f_{n_{k+1}} - f_{n_k})| \le G$ a.e., the [Dominated Convergence Theorem](/theorems/511) gives $\|f_{n_k} - f\|_p \to 0$. Finally, $\|f\|_p \le \|f - f_{n_1}\|_p + \|f_{n_1}\|_p < \infty$, so $f \in L^p$. Since $(f_k)_k$ is Cauchy and the subsequence $(f_{n_k})_k$ converges to $f$, the full [sequence](/page/Sequence) converges to $f$. **Case 2: $p = \infty$.** Let $(f_n)_n$ be Cauchy in $L^\infty$. The [set](/page/Set) $A_{m,n} := \{x : |f_n(x) - f_m(x)| > \|f_n - f_m\|_\infty\}$ has measure zero for each $m, n$. The union $A := \bigcup_{m,n} A_{m,n}$ has measure zero, and on $E \setminus A$ the sequence $(f_n)_n$ is uniformly Cauchy. Hence $f_n \to f$ uniformly on $E \setminus A$ for some bounded [function](/page/Function) $f$. Extending $f$ by $0$ on $A$ gives $f \in L^\infty$ with $\|f_n - f\|_\infty \to 0$.