$In$ this question, $we$ prove one $of$ the fundamental properties $of$ the integral.

$(a)$ Let $f,g$:$[a,b]Rbe$ two bounded functions. Prove that

$su{p}_{\text{x}\text{i}\text{n}[a,b]}(f(x)+g(x))su{p}_{\text{x}\text{i}\text{n}[a,b]}f(x)+su{p}_{\text{x}\text{i}\text{n}[a,b]}g(x),$

$in{f}_{\text{x}\text{i}\text{n}[a,b]}(f(x)+g(x))in{f}_{\text{x}\text{i}\text{n}[a,b]}f(x)+in{f}_{\text{x}\text{i}\text{n}[a,b]}g(x).$

$(b)$ Use part $(a)to$ show that for any partition $Pofa,b,we$ have

$U_P(f+g)U_P(f)+U_P(g)$;$,L_P(f+g)L_P(f)+L_P(g).$

$(c)$ Use part $(b)to$ prove that $iff$ and $g$ are integrable, then

${\displaystyle \underset{a}{\overset{b}{}}}f(x)+g(x)dx={\displaystyle \underset{a}{\overset{b}{}}}f(x)dx+{\displaystyle \underset{a}{\overset{b}{}}}g(x)dx$