> [!NOTE] Theorem > Let $f,g:[a,\infty)\to \mathbb{R}$ be [[Real Function|real functions]] that are [[Riemann integration|integrable]] on each interval $[a,b]$ for all $a\leq b.$ If for all $x\geq a,$ $|f(x))|\leq g(x)$ and the [[Improper Integral Over Unbounded Closed Interval|improper integral]] $\int_{a}^{\infty} g(x) \, dx$ exists, then $\int_{a}^{\infty} f(x) \, dx$ also exists. **Proof**: Let $b\geq a.$ We have that the functions $|f|$ and $f+|f|$ are integrable on each interval $[a,b].$ By [[Monotonicity of Riemann Integral]], $\int_{a}^{b} |f(x)| \, dx \leq \int_{a}^{b} g(x) \, dx \leq \int_{a}^{\infty} g(x) \, dx $and $\int_{a}^{b} (f(x)+|f(x)|) \, dx \leq 2 \int_{a}^{b} g(x) \, dx \leq 2 \int_{a}^{\infty} g(x) \, dx $Both functions $|f|$ and $f+|f|$ are non-negative so the functions $b \mapsto \int_{a}^{b} |f(x)| \, dx $and $b \mapsto \int_{a}^{b} (f(x)+|f(x)|) \, dx$ are bounded increasing functions. So the limits of both as $x\to \infty$ exists and hence by [[Sum Rule for Limits of Real Functions]] so does their difference $b \mapsto \int_{a}^{b} f(x) \, dx. $