Exponential bound for a lower-sum product
= Exponential bound for a lower-sum product
For a nonnegative <Riemann-integrable function> and an interval partition,
$$
\prod_k(1+m_k\Delta x_k)\leq e^{\sum_km_k\Delta x_k}\leq e^{\int_a^bf(x)\,dx},\qquad m_k=\inf_{[x_{k-1},x_k]}f.
$$
Every factor is nonnegative, so multiplication of $1+t\leq e^t$ is legitimate. The final step uses the lower-sum bound on the <Riemann integral>.