Suppose that $\displaystyle f:[a,b] \rightarrow \mathbb{R}$ is a Riemann integrable on $\displaystyle [a,b]$ and $\displaystyle f(x) \geq 0 $ for all $\displaystyle x \in [a,b]$. Prove that $\displaystyle \sqrt{f}$ is Riemann integrable on $\displaystyle [a,b]$.

Relevant Theorems & Definitions

Definition - Riemann integrable - if upper integral of $\displaystyle f(x)dx$= lower integral of $\displaystyle f(x)dx$.

Theorem - Riemann integrable iff $\displaystyle \exists A \in \mathbb{R}$ such that $\displaystyle \forall \epsilon >0 \exists$ a partition $\displaystyle P$ of $\displaystyle [a,b]$ such that $\displaystyle \forall$ marked refinements $\displaystyle Q$ of $\displaystyle P$, $\displaystyle |S(Q, f) - A | \leq \epsilon$, where $\displaystyle A=\int^a_b f(x)dx$.

Theorem - Let $\displaystyle f: [a,b] \rightarrow \mathbb{R}$ be a continuous function on $\displaystyle [a,b]$. Then $\displaystyle f$ is Riemann integrable on $\displaystyle [a,b]$.

etc.

Theorem - Riemann integrable iff for each $\displaystyle \epsilon > 0 \exists$ a partition $\displaystyle P$ of $\displaystyle [a,b]$ such that $\displaystyle U(P,f)-L(P,f) \leq \epsilon$.

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