Lebl-Basic-Analysis-I · Problem 5.2.2

The Riemann Integral

Mathematics · linearity-of-integral

Problem

Prove the second part of Proposition 5.2.4. Let

f

and

g

be in

\mathcal{R}([a,b])

. Prove, without using Proposition 5.2.5, that

f + g

is in

\mathcal{R}([a,b])

and

\int_a^b \bigl(f(x) + g(x)\bigr)\,dx = \int_a^b f(x)\,dx + \int_a^b g(x)\,dx.

Hint: One way to do it is to use Proposition 5.1.7 to find a single partition

P

such that

U(P,f) - L(P,f) < \epsilon/2

and

U(P,g) - L(P,g) < \epsilon/2