TITLE: Prove that ${1\over2}<{1\over1001}+......+{1\over2000}<1$
QUESTION [7 upvotes]: Prove that ${1\over2}<{1\over1001}+......+{1\over2000}<1$
Can it be proved by langrange's mean value theorem or by convert it into a Riemann sum?

REPLY [14 votes]: $$\frac{1}{2}=\frac{1000}{2000}<\underbrace{\frac{1}{1001}+\cdots+\frac{1}{2000}}_{1000 \text{  terms}}<\frac{1000}{1000}=1$$