Abstract
In this paper, by considering usual partition of \([0, \infty)\) \(\Delta\)-convergence of non-negative real valued sequences is defined. It is shown that every convergent sequence is \(\Delta\)-convergence but the converse is not true, in general. Besides, some basic properties of \(\Delta\)-convergence as well as the second part of this paper by using any lacunary sequences as a partition of non-negative real numbers, lacunary uniform distribution is defined and some inclusion result between uniform distribution modulo 1 and lacunary uniform distribution has been given.
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