Proposition 4 of Chapter $4$ From Royden Real Analysis

Question

From Real Analysis By Halsey Royden,

Please someone let me know,how to show

$R \underline{\int_{a}^{b}}f(x)dx \leq sup_{\phi \leq f}\int_a^b \phi(x) dx$

I was trying using the definition given

Thanks in advance!

Answer 1

To show the result you want, we first show that every lower Riemann sum (called $s$ in the textbook, is less than or equal to $\sup_{\phi\leq f}\int_a^b \phi(x)dx$. Looking at the definition of $s$, you have $s=\int_a^b \varphi(x)dx$ for some step function $\varphi$ defined such that $\varphi(x)=m_i$ on $(\xi_{i-1},\xi_i]$. It can be seen that $\varphi\leq f$, and thus $$s=\int_a^b \varphi(x)dx\leq \sup_{\phi\leq f}\int_a^b \phi(x)dx$$ Taking supremum over $s$ you get the desired result.