1. ## Integral Theorem

Theorem. Let $f$ and $g$ be real valued functions that are Riemann integrable on $[a,b]$. Then the following statements hold:

1. The function $f+g$ is Riemann integrable on $[a,b]$ and $\int_{a}^{b} (f+g) = \int_{a}^{b} f + \int_{a}^{b} g$.

2. The function $kf$ is Riemann integrable on the interval $[a,b]$ and $\int_{a}^{b} kf = k \int_{a}^{b} f$.

3. The function $f-g$ is Riemann integrable on $[a,b]$ and $\int_{a}^{b} (f-g) = \int_{a}^{b} f - \int_{a}^{b} g$.

Outline of Proof. The basic idea is that we basically look at $\mathcal{R}(f+g,P), \ \mathcal{R}(kf, P),$ and $\mathcal{R}(f-g, P)$ and show that they are in fact equal to those right hand sides?

2. Originally Posted by manjohn12
1. The function $f+g$ is Riemann integrable on $[a,b]$ and $\int_{a}^{b} (f+g) = \int_{a}^{b} f + \int_{a}^{b} g$.
Let $I_1 = \smallint_a^b f \text{ and }I_2 = \smallint_a^b g$. For $\epsilon > 0$ there exists $\delta > 0$ so that $| \mathcal{R}(f,P) - I_1| < \frac{\epsilon}{2}$ and $|\mathcal{R}(g,P) - I_2| < \frac{\epsilon}{2}$ as long as $||P|| < \delta$.

Now, $\mathcal{R}(f+g,P) = \mathcal{R}(f,P) + \mathcal{R}(g,P)$.
Thus, $|\mathcal{R}(f+g,P) - I_1 - I_2| \leq |\mathcal{R}(f,P) - I_1| + | \mathcal{R}(f,P) - I_2| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon$.

EDIT: Fixed

3. Originally Posted by ThePerfectHacker
Let $I_1 = \smallint_a^b f \text{ and }I_2 = \smallint_a^b g$. For $\epsilon > 0$ there exists $\delta > 0$ so that $| \mathcal{R}(f,P) - I_1| < \frac{\epsilon}{2}$ and $|\mathcal{R}(g,P) - I_2| < \frac{\epsilon}{2}$ as long as $||P|| < \delta$.

Now, $\mathcal{R}(f+g,P) = \mathcal{R}(f,P) + \mathcal{R}(f,P)$.
Thus, $|\mathcal{R}(f+g,P) - I_1 - I_2| \leq |\mathcal{R}(f,P) - I_1| + | \mathcal{R}(f,P) - I_2| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon$.
$\mathcal{R}(f+g,P) = \mathcal{R}(f,P) + \mathcal{R}(g,P)$.

Thus, $|\mathcal{R}(f+g,P) - I_1 - I_2| \leq |\mathcal{R}(f,P) - I_1| + | \mathcal{R}(g,P) - I_2| < \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon$.