Prove or disprove this condition for convergence of a sequence Let $a_n$ be a sequence and suppose we have the condition
$a_{n+f(n)}-a_n\rightarrow 0$ is true for any $f:\mathbb{N}\rightarrow\mathbb{N}$. 
Does it hold that $a_n$ is convergent?
So I have solved a similiar problem when the condition is that $f(n) = k$ for all value of $k$. In this case, it is not sufficient, because we may take the sequence $\sqrt{n}$ or the partial sum of harmonic series, but here I am clueless. Any ideas?

HINT: Suppose that $\langle a_n:n\in\Bbb N\rangle$ is not convergent. If it has an unbounded subsequence, then it’s easy to find an $f:\Bbb N\to\Bbb N$ such that $a_{n+f(n)}-a_n\not\to 0$, so assume that $\langle a_n:n\in\Bbb N\rangle$ is bounded but not convergent. Then the sequence has two subsequences converging to different limits, say $L$ and $M$. Use these two subsequences to construct a function $f:\Bbb N\to\Bbb N$ and a subsequence $\langle a_{n_k}:k\in\Bbb N\rangle$ such that $a_{n_k+f(n_k)}-a_{n_k}\to L-M\ne 0$ as $k\to\infty$.
