This is the problem: $$ \frac n2 > \sum_{k=1}^n (1/k) -1$$
Here's my try.
P(1): $$\frac12 > \frac11 -1$$
$$\frac12 > 0$$
P(n) => P(n+1): $$\sum_{k=1}^{n+1} \frac1k - 1 = \sum_{k=1}^n \frac1k + \frac 1{n+1} - 1 < \frac n2 + \frac 1{n+1}$$
Now the right side of inequality is bigger than sum to $n+1$ which is bigger than sum to $n$ that means that it's bigger than sum to $n$.
We have $$\frac n2+\frac1{n+1} > \sum_{k=1}^n \frac1k - 1$$
I don't know what to do next. I wrote this above to help you to help me with this kind of problems. I get confused when i need to prove inequality using induction.
Thanks!