How to prove the sequence $\{b_n\}$ with $0 \leq b_{n+1} \leq b_n + a_n$ converges, where $a_n \geq 0$ converges to $0$.

Question

Let $\{a_n\}$ be a sequence of non-negative real numbers that converges to zero. Suppose the sequence $\{b_n\}$ of non-negative real numbers satisfies \begin{equation} 0 \leq b_{n+1} \leq b_n + a_n \end{equation} for all $n \in \mathbb{N}$. Is it possible to prove that $\{b_n\}$ also converges? The only thing I found is that if $\sum_{m=1}^\infty a_n < \infty$, then $b_n$ is bounded as follows. \begin{equation} 0 \leq b_{n+1} \leq b_n + a_n \leq b_{n-1} + a_n + a_{n-1} \leq \cdots \leq b_1 + \sum_{m=1}^n a_m \leq b_1 + \sum_{m=1}^\infty a_m < \infty \end{equation} Is it possible to prove the boundedness and convergence of $b_n$ without assuming $\sum_{m=1}^\infty a_n < \infty$?

Answer 1

Answer : NO. Counterexample : take $a_n=\frac{1}{n}$, $b_n=\sum_{k=1}^{n-1} a_k$