A challenging question on Riemann integration ( Reference- Real Analysis/A.J.White) If $f:[a,b]→R$ is Riemann integrable over [a,b] and $f≥0$, then given  $ϵ&gt;0$ there exists $δ&gt;0$ such that if $P=(y_1,...,y_m)$ (which is a partition of [a.b]) and max{$y_{s+1}$-$y_{s}$:s=1,...,m-1}&lt;$δ$, then $U(P,f)-\int_{a}^{b}f&lt;ϵ$.
$$$$I would much appreciate if someone could help me out with this question. Thanks
Complete answer: $[L\leqslant I\leqslant U\land U-L\leqslant\varepsilon]\implies U-I\leqslant\varepsilon$.
Here is the answer I figured out and I am putting it here.  Let ε>0 be arbitrary. Then there exists a partition $P_1$={$x_0$,...,$x_p$} in  $[a,b]$ such that $U(P_1,f)&lt;\int_{a}^{b}f+ϵ/2$ 
 Choose $δ=ϵ/4K(p-1)$ where K is the upper bound of $f(x)$.
Let P be any partition of $[a,b]$ with $||P||&lt;δ$.

Let $P'$ be the union of $P$ and $P_1$ .

$P'$ has at most $(p-1)$ points that $P$. $$U(P,f)-2K(p-1)δ⩽U(P',f)⩽U(P_1,f)&lt;\int_{a}^{b}f+ϵ/2$$ Thus,

$$ U(P,f)&lt;\int_{a}^{b}f+ϵ$$ for any partition with $||P||&lt;δ$.