Let $f_{1}$ and $f_{2}$ be Riemann integrable functions on $[a, b]$. Show that $f_{1}+f_{2}$ is Riemann integrable.

Let $f$ be a Riemann integrable function on $[a, b]$ and set $f^{+}(x)=\max (f(x), 0)$. Show that $f^{+}$and $|f|$ are Riemann integrable.

Let $f$ be a function on $[a, b]$ such that $|f|$ is Riemann integrable. Is it true that $f$ is Riemann integrable? Justify your answer.

Show that if $f_{1}$ and $f_{2}$ are Riemann integrable on $[a, b]$, then so is $\max \left(f_{1}, f_{2}\right)$. Suppose now $f_{1}, f_{2}, \ldots$ is a sequence of Riemann integrable functions on $[a, b]$ and $f(x)=\sup _{n} f_{n}(x)$; is it true that $f$ is Riemann integrable? Justify your answer.