Hi could anyone explain me how one come sup with the term circled with red? The proof tries to show that for integer numbers the mimimum is somewhere else (a bit higher) than it would be if f would be continous ($k_{0}$ is where the minimum is btw). Thanks!
The first inequality probably comes from the Taylor expansion of $\ln(f)$ about $k_0$, since you're told the second derivative.
Since $k_1$ is the integer closest to the right of $k_0$ the difference between them is less than $1$ so the square of the difference is less than $1$. Then omitting that factor strengthens the inequality.