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?

Well, let's proceed with the same sort of approach you tried, but a bit finer.
There are $100$ terms greater than $1/200$, another $100$ terms greater than $1/300$, and so on, up to $100$ terms greater than $1/1000$. But this gives that the sum is greater than $1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + 1/10$, which is about $1.9$, but more importantly is very easily calculatable to be greater than $1$.
