<s>[INST] Which of these statements about $Q=\frac{1}{100}+\frac{1}{101}+\cdots+\frac{1}{1000}$ is correct? Pick the correct option regarding $Q$.

$$Q=\frac{1}{100}+\frac{1}{101}+\cdots+\frac{1}{1000}$$
  Pick one option:
  
  
*
  
*$Q>1\qquad$ 2. $Q\leq \frac{1}{3}\qquad$ 3. $\frac{1}{3}<Q\leq \frac{2}{3}\qquad$ 4. $\frac{2}{3}<Q\leq 1$
  

My approach :
$$Q = \frac{1}{100}+\frac{1}{101}+\cdots+\frac{1}{1000} > \frac{1}{1000}+\frac{1}{1000}+\cdots+\frac{1}{1000} = \frac{901}{1000}$$
$$\implies \frac{1}{100}+\frac{1}{101}+\cdots+\frac{1}{1000} > \frac{9}{10}$$
So, $Q > 1$. Option 1 is correct.
Now, my question is: can this be proved with some other approach?
 [/INST] Divide the sum into two parts:
$$\frac1{100}+\ldots+\frac{1}{500}+\frac1{501}+\ldots+\frac1{1000}$$
The first part, $100$ to $500$ is such that:
$$\frac1{100}+\ldots+\frac{1}{500}>\frac1{500}+\ldots+\frac{1}{500}=\frac{401}{500}$$
The second part, $501$ to $1000$ is such that:
$$\frac1{501}+\ldots+\frac{1}{1000}>\frac1{1000}+\ldots+\frac{1}{1000}=\frac{500}{1000}$$
Therefore the sum of $100$ to $1000$ is larger than $\frac{401}{500}+\frac{500}{1000}$, which itself is larger than $1$.
</s>