Problem 2343

We say that a strictly increasing sequence of positive integers $n_1$, $n_2$, $\ldots$ is non-decelerating if $n_{k+1}-n_k\le n_{k+2}-n_{k+1}$ holds for all positive integers $k$. We say that a strictly increasing sequence $n_1$, $n_2$, $\ldots$ is convergence-inducing, if the following statement is true for all real sequences $a_1$, $a_2$, $\ldots$: if subsequence $a_{m+n_1}$, $a_{m+n_2}$, $\ldots$ is convergent and tends to 0 for all positive integers $m$, then sequence $a_1$, $a_2$, $\ldots$ is also convergent and tends to 0. Prove that a non-decelerating sequence $n_1$, $n_2$, $\ldots$ is convergence-inducing if and only if sequence $n_2-n_1$, $n_3-n_2$, $\ldots$ is bounded from above.