$N \in \mathbb N$.
$\displaystyle\int_{N-1}^N \left(\dfrac{1}{x} - \dfrac{1}{N}\right) dx>0$
This is the finish of a proof, a modification of $\log N-\log (N-1) -\frac{1}{N}$. Calculating it out wouldn't be a problem, but how can I "see" that this assertion is valid?
Is it because $1/N$ is always smaller or equal to $1/x$?
Since on $[N-1,N]$ you have $\frac1x \geq \frac1N$, the function you integrate is non-negative. It is also continuous; the two together imply that the integral is strictly positive, as the integral of a non-negative continuous function is zero iff the function is zero on the whole integration domain (which is not the case, e.g. looking at the middle point of the interval).