Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 1 / 10F (Analysis I)

Pointwise on $[a,b]$,because $g(x)>0$. To see the implication directly from the definition, every Riemann sum of the nonnegative continuous function $(f-m)g$ is nonnegative, so its limit is nonnegative:Applying the same argument to $(M-f)g$ gives ${\int }_a^b(M-f)g\ge 0$. By linearity, these are exactly the monotonicity of the Riemann integral bounds

LetContinuity at zero gives $m_n,M_n\to f(0)$. Part (a), with the positive weight $n{e}^{-nx}$, givesThe squeeze theorem therefore yields

Since $f$ is bounded on $[0,1]$, say $|f|\le K$, the omitted tail satisfiesAdding the tail proves