How to prove the fraction identity without using calculator

Question

How to prove without calculator that

$$ \frac{1}{1001} + \frac{1}{3001} > \frac{1}{1000}$$

Answer 1

$$\frac{1}{1001}+\frac{1}{3001}=\frac{3}{3003}+\frac{1}{3001}>\frac{3}{3003}+\frac{1}{3003}=\frac{4}{3003}>\frac{4}{4000}=\frac{1}{1000}$$

Answer 2

HINT:

$$\frac1{1000}-\frac1{1001}=\frac{1001-1000}{1000\cdot1001}=\frac1{1000\cdot1001}<\frac1{3001}$$