$(i)$-Convergence and its application to a sequence of functions

Let $(x_\alpha)_{\alpha\in A}$ , where $A$ is a directed set containing cofinal chains — a generalized sequence in a complete chain. It is established that every such sequence contains a monotonic cofinal sub-sequence. For a monotonically increasing (decreasing) bounded sequence $(x_\alpha)_{\alpha\in A}$, by definition, we put $(i)-\lim\limits_{\alpha\in A}x_\alpha=\sup\limits_{\alpha\in A}(x_\alpha)\cdot((i)-\lim\limits_{\alpha\in A}x_\alpha=\inf\limits_{\alpha\in A}(x_\alpha))$. For an arbitrary sequence $(x_\alpha)_\alpha\in A(i)$ the $(i)$-limit is defined as the common $(i)$-limit of its monotonic cofinal sub-sequences. The properties of $(i)$-convergence and some of its applications to generalized sequences of mappings are discussed.