# Prove that ${1\over2}<{1\over1001}+…+{1\over2000}<1$

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?

• Hint: You're vastly overkilling it. – Git Gud Jun 6 '14 at 11:39
• Why proving that a finite sum is in $(0.5,1)$? Just calculate the value and you're done. – Thekwasti Jun 6 '14 at 11:42
• – OBDA Jun 6 '14 at 11:56

$$\frac{1}{2}=\frac{1000}{2000}<\underbrace{\frac{1}{1001}+\cdots+\frac{1}{2000}}_{1000 \text{ terms}}<\frac{1000}{1000}=1$$