How to prove the inequality $\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n-1}\geq \log (2)$?

Use Riemann sum. Observe \begin{align} \frac{1}{n} + \ldots + \frac{1}{2n-1}\geq \int^{2n}_{n} \frac{1}{x}\ dx = \log 2n-\log n = \log 2. \end{align}

It's well known that the following inequality holds: $$e^x\geqslant x+1\geqslant 0,\ \forall\ x\geqslant 0.$$ So we have $$\displaystyle\prod_{i=n}^{2n-1}e^{{1\over i}}\geqslant \prod_{i=n}^{2n-1}\big(1+{1\over i}\big)=\prod_{i=n}^{2n-1}{i+1\over i}={2n\over n}=2.$$ Thus $$\displaystyle\sum_{i=n}^{2n-1}{1\over i}\geqslant \ln2.$$