# Sum of Riemann integrable functions

I am trying to study real analysis on my own and I need help proving that the sum of two riemann integrable functions is also riemann integrable. The exact problem statement from the book:

Let $$f$$ and $$g$$ be bounded, Riemann integrable functions on $$[a,b]$$. Show that $$f+g$$ is Riemann integrable on $$[a,b]$$ with $$\int\limits_a^b(f+g)=\int\limits_a^bf+\int\limits_a^bg$$.

My attempt at the solution so far:

I am trying to prove this using the integrability criterion $$U(f+g,P)-L(f+g,P)<\epsilon$$, where $$U$$ and $$L$$ are the upper and lower sums of $$f+g$$ with respect to $$P$$.

Let $$P_f$$ and $$P_g$$ be partitions of $$[a,b]$$ that satisfy the condition. Fix $$\epsilon >0$$. By the integrability of $$f$$ and $$g$$:

$$U(f,P_f)-L(f,P_f)<\frac{\epsilon}{2}$$

$$U(f,P_f)-L(f,P_f)<\frac{\epsilon}{2}$$

$$U(g,P_g)-L(g,P_g)<\frac{\epsilon}{2}$$

Let $$P = P_f\bigcup P_g$$. Since $$P$$ is a refinement of both $$P_f$$ and $$P_g$$ we have

$$L(f,P_f)\leq L(f,P)$$, $$U(f,P)\leq U(f,P_f)$$, $$L(g,P_g)\leq L(g,P)$$, $$U(g,P)\leq U(g,P)$$

Therefore,

$$U(f,P)-L(f,P)<\frac{\epsilon}{2}$$

$$U(g,P)-L(g,P)<\frac{\epsilon}{2}$$

Now I am stuck because I can't establish a relation between the upper sum of each function individually and the upper sum of both functions with respect to $$P$$ (Same for lower sums). Any advice?

• The answer below is fine, but if you want an answer which follows your idea, let me know – Matematleta Oct 19 '18 at 23:50

Let $$P$$ be any partition of $$[a,b]$$. If $$J$$ is an interval in the partition then it is easy to see that $$\sup_J(f+g)\leq \sup_Jf+\sup_Jg$$ and $$\inf_J(f+g)\geq \inf_Jf+\inf_Jg$$. So if the partition $$P$$ is $$a=x_0 then:
$$U(f+g,P)=\sum_{i=1}^n \sup_{[x_{i-1},x_i]}(f+g)\times (\Delta x_i)\leq \sum_{i=1}^n \sup_{[x_{i-1},x_i]}(f)\times (\Delta x_i)+\sum_{i=1}^n \sup_{[x_{i-1},x_i]}(g)\times (\Delta x_i)=U(f,P)+U(g,P)$$
$$L(f+g,P)=\sum_{i=1}^n \inf_{[x_{i-1},x_i]}(f+g)\times (\Delta x_i)\geq \sum_{i=1}^n \inf_{[x_{i-1},x_i]}(f)\times (\Delta x_i)+\sum_{i=1}^n \inf_{[x_{i-1},x_i]}(g)\times (\Delta x_i)=L(f,P)+L(g,P)$$