Assume that a sequence $(x_n)_{n\in\omega}$ of points of a locally convex topological vector space converges to zero. Is it always possible to find increasing number sequences $(n_k)_{k\in\omega}$ and $(m_k)_{k\in\omega}$ such that the sequence $(m_kx_{n_k})_{k\in\omega}$ still converges to zero?

**Added in Edit.** So we already know that this property (called *the Mackey convergence condition*) does not hold in any locally convex space. But we can ask another

**Problem.** Assume that a locally convex space $X$ admits an indexed family $(B_\alpha)_{\alpha\in\omega^\omega}$ of bounded sets such that (i) $B_\alpha\subset B_\beta$ for all $\alpha\le \beta$ in $\omega^\omega$ and (ii) each bounded subset $B\subset X$ is contained in some $B_\alpha$, $\alpha\in\omega^\omega$.

Does $X$ satisfy the Mackey convergence condition?