1. ## Riemann integrable, function

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 $\displaystyle U(P,f)-L(P,f) \leq \epsilon$.

I haven't gotten very far doing this problem. I need help with this problem. Thank you.

2. This can be more "easily" solved if you work with the following.
Suppose $\displaystyle f$ is Riemann integrable, and $\displaystyle g$ continuous, on $\displaystyle [a,b]$. Then $\displaystyle g\circ f$ is Riemann integrable on $\displaystyle [a,b]$.

Hint to proof: First consider a partition $\displaystyle P$ such that $\displaystyle U(P.f)-L(P.f)<\delta^2$. Then make two sets as follows; $\displaystyle A:=\{i:M_i-m_i<\delta\}$ and $\displaystyle B:=A^c$. You choose $\displaystyle \delta$ needed for the definition of continuity for $\displaystyle g$. Also since $\displaystyle f$ is Riemann integrable, it is bounded on this interval (you will need this).