I came up with the following result from Xu 20212, An iterative approach to quadratic optimization, J. Optim. Theory Appl 116 (2003) 659-678. Let $(s_n)$ be a sequence of nonnegative numbers satisfying the condition
$$ s_{n+1}\le (1-\alpha_n)s_n+\alpha_n\beta_n,n\ge 0$$ where $(\alpha_n),(\beta_n)$ are sequences of real number such that
(i) $(\alpha_n)\subset[0,1]$ and $\sum_{n=0}^\infty \alpha_n=\infty$.
(ii) $\limsup_{n\rightarrow\infty}\beta_n\le 0$ or
(ii')$\sum_n\alpha_n\beta_n$ is convergent.
Then $\lim_{n\rightarrow \infty}s_n=0$.
I wonder if there are any other references for special cases of this result. I found several here Limit of a sequence and Assuming $0 \leq a_{n+1} \leq c_n a_n + b_n$ (+ other conditions), show $a_n \to 0$. I also remember that there is a Russian book for undergraduate students discussing these types of lemmas, but now I couldn't find it again.
Thanks a lot for any help, comment or discussion.