TITLE: Riemann Integrable Summation of Functions
QUESTION [1 upvotes]: Let $f$ and $g$ be Riemann Integrable. Prove $f+g$ is Riemann Integrable and
$$\int_a^bf(x)+g(x)dx= \int_a^b f(x)dx + \int_a^bg(x)dx$$
I have the following:
For a partition $P$ of $[a,b]$
$L(p;f+g)=$$$\sum_{i=1}^{n} \min(f+g)\Delta x_i$$ $$\ge \sum_{i=1}^{n}\min(f)\Delta x_i + \sum_{i=1}^{n}\min(g)\Delta x_i$$
$$=L(p;f)+L(p;g)$$ 
Which then implies the equalities of the summations of upper and lower Darboux Integrals.
The only part I need help with is how to show the inequality with the summations.

REPLY [1 votes]: Clearly $$f(x)+g(x)\leq f(x)+g(x)$$ for all $x\in[a_i,b_i]$. Then, from the definitions, $$\inf_{x\in[a_i,b_i]}f(x)+\inf_{x\in[a_i,b_i]}g(x)\leq\inf_{x\in[a_i,b_i]}(f(x))+g(x)\leq f(x)+g(x).$$This shows $\inf_{x\in[a_i,b_i]}f(x)+\inf_{x\in[a_i,b_i]}g(x)$ is a lower bound for $f(x)+g(x)$, so the infimum cannot be less than that since the infimum is, by definition, the greatest lower bound. Thus, $$\inf_{x\in[a_i,b_i]}f(x)+\inf_{x\in[a_i,b_i]}g(x)\leq\inf_{x\in[a_i,b_i]}(f(x)+g(x)).$$
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