I need a function $f(x)$ that satisfies the properties bellow for all integers $k$
$ \frac{\log(k+1)}{k+1}-\log\left(1+\frac 1 k\right)+f(k+1)-f(k)<0 \ $ $ \lim_{k \rightarrow \infty} f(k)=0 $
I don't think it should be very hard sense if I let $f(x)=0$, the entire thing is already very close to zero. In addition if you find a function that doesn't work for the first few values 1,2,3.. etc, thats fine too. I would appreciate any help, thanks.