From H.L.Royden's Real Analysis,
Here They haven't shown how to prove $cf$ and $f+g$ is integrable.
The definition of integral and proposition $8$ given as: 
Using this my proof of $\int_E cf=c\int_E f$ :
\begin{eqnarray*}\int_E cf &=& \sup_{h \leq cf}\int_E h \\&=& \sup_{\frac{h}{c} \leq f}\int_E h \\ &=& \sup_{h^{'} \leq f}\int_E ch'(letting \frac{h}{c} = h^{\prime}) \\&=& \sup_{h^{\prime} \leq f} c\int_E h\\ &=&c\int_E f \\ \end{eqnarray*} is this correct?
Second thing Tell me how to show $cf$ is integrable.
Definition given:A measurable function $f$ is said to be integrable over $E$ if $f^+$ and $f^-$ are both integrable over E.
I was trying show $(cf)^+$ and $(cf)^-$ is integrable.$f$ is integrable implies $f^+$ and $f^-$ is integrable.But I am uanble to use this to show the required..
The way such a proof goes is :
If $f$ is a simple function : $f=\sum_{i=1}^n a_i1_{A_i}$ where $A_i$ are measurable. Then $$\int cf=\sum_{i=1}^nca_im(A_i)=c\sum_{i=1}^n a_im(A_i)=c\int f.$$
If $f\geq 0$, then there is a sequence of simple function s.t. $f_n\nearrow f$. Then, $$c\int f=c\lim_{n\to \infty }\int f_n=\int cf_n=\int cf,$$
where the last equality comes from the fact that $(cf_n)$ is a sequence of simple function s.t. $cf_n\nearrow cf.$
Finally, if $f$ is measurable, then $f=f^+-f^-$ and $cf=cf^+-cf^-=(cf)^+-(cf)^-$, and use the previous step.