simple induction, why does it go this way? I have a simple question involving induction. 
$$\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n}\geq \frac{1}{2}.$$
I think that this does not go with most common version of induction, because of the $\frac{1}{n}$ at the beginning (am I right?). So here is a need to add something to the right side of inequality and then check whether the statement is still true (how do I guess the term which is to be added?). Mate suggested to add to $\frac{1}{2}$ the  $\frac{1}{2n-1}$ term. But why this? Thank you in advance for time and answer.

$$\frac{1}{n}+\frac{1}{n+1}+\ldots +\frac{1}{2n}\geq \frac{1}{2n}+\frac{1}{2n}+\ldots +\frac{1}{2n}=\frac{n+1}{2n}\ge\frac{n}{2n}=\frac{1}{2}$$
