Let\((x_\alpha )_{\alpha \epsilon A}\), 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\((x_\alpha )_{\alpha \epsilon A}\), by definition, we put\((i) - \mathop {\lim }\limits_{\alpha \in A} x_\alpha = \mathop {\sup }\limits_{\alpha \in A} (x_\alpha ) \cdot ((i) - \mathop {\lim }\limits_{\alpha \in A} x_\alpha = \mathop {\inf }\limits_{\alpha \in A} (x_\alpha ))\). For an arbitrary sequence\((x_\alpha )_{\alpha \epsilon A}\) 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.