On a notion of asymptotically monotone sequence

Let us say that an infinite sequence $\{a_n\}$ of real numbers is asymptotically increasing if there exists a double sequence$\{b_{n,m}\}_{n<m}$ such that

(1) $b_{n,m}\rightarrow 0$ as $n,m \rightarrow \infty$, that is, for every $\varepsilon>0$ there is $N$ such that for all $m>n>N$ one has $|b_{n,m}|<\varepsilon$.

(2) $a_n\leq (1+b_{n,m})a_m$ whenever $n<m$.

(I don't know whether this is a standard definition, but that's the definition that suits my purposes.) So, when all $b_{n,m}$ are zero, we get the definition of (non-strictly) increasing sequence.

My question is: Is it true that an asymptotically increasing sequence (in the above sense) which is bounded is convergent?