<s>[INST] Showing that $S=\frac{1}{100} + \frac{1}{101} + \dots + \frac{1}{1000} \gt 1$ 
If $$S=\frac{1}{100} + \frac{1}{101} + \dots + \frac{1}{1000}$$ then
  $$S\gt 1,$$

but how?
I understood that there are $451$ pair of terms. So clubbed two terms together.
$\frac{1}{100}+\frac{1}{1000}+\frac{1}{101}+\frac{1}{999}.....$
But I am not able to solve it further. Such a tricky question for me.
 [/INST] \begin{eqnarray*}
\frac{1}{129}+\cdots + \frac{1}{256} > \frac{1}{2} \\
\frac{1}{257}+\cdots + \frac{1}{512} > \frac{1}{2} \\
\end{eqnarray*}
</s>