Does there exist a real function $f$ with the following properties?
(a) $f$ is positive function ($f\geq 0$).
(a) $f$ is concave and increasing on $[1, +\infty)$.
(b) $\displaystyle{\lim_{n\to\infty}}(f(n) - f(n+1)) = 0$.
(c) $\displaystyle{\lim_{x\to\infty}}f(x)\leq 0$.
Anyone can help me. Thanks
At the time of writing this answer, the four conditions in your question are
(a) $f$ is positive function ($f\geq 0$).
(a) $f$ is concave and increasing on $[1, +\infty)$.
(b) $\displaystyle{\lim_{n\to\infty}}(f(n) - f(n+1)) = 0$.
(c) $\displaystyle{\lim_{x\to\infty}}f(x)\leq 0$.
A monotonically increasing function is smaller or equal to its limit when $x\to\infty$. If $\lim_{x\to\infty} f(x)\le 0$ (from (c)), and $f$ is increasing (from (a)), then it must hold that
$$f(x)\le 0\quad\mathrm{for}\ x\in[1,\infty).$$
But from the other (a), $f(x)\ge 0$ for all $x$. The only function that satisfies simultaneously both conditions is
$$f(x)=\begin{cases}0, & \mathrm{if}\ x\in[1,\infty) \\ \textrm{anything positive,} & \mathrm{if}\ x<1\end{cases}$$