For a given real sequence $\{a_k\} \subset \mathbb{R}^n$, suppose the sequence satisfies relation $| a_k - a_{k-1} | \le \frac 1 k$. We know this does not guarantee the sequence to be Cauchy. If we take $a_k = \sum_{j=1}^k \frac 1 j$, the sequence is apparently divergent.
I am wondering if we know a priori that the sequence converges to a limit point $\lim_{k \to \infty} a_k = a$. Would the relation $|a_k - a_{k-1} | \le \frac 1 k$ provide any interesting information about the sequence? For example, the rate of convergence.
Even if the limit of $a_k$ does not exist (diverges to $\pm\infty$ or oscillates) this would tell you that with convergence rate no slower than $1/k$
$$\lim_{k \to \infty} \frac{a_k}{k} = 0$$.
You can prove with Cesaro-Stolz theorem:
$$\lim_{k \to \infty} \frac{a_k}{k} = \lim_{k \to \infty} \frac{a_k - a_{k-1}}{k - (k-1)} = \lim_{k \to \infty}(a_k - a_{k-1}) = 0$$