INTERNATIONAL JOURNAL OF ADVANCE RESEARCH
IN MULTIDISCIPLINARY

Using the result ‘every sequence of real numbers has a monotone subsequence’ we prove: If a bounded sequence has unequal lim sup and lim inf, then it has at least two monotone (convergent) subsequences whose range sets are disjoint; (BW theorem) Every bounded sequence has a convergent subsequence, and R is a complete space under usual metric on R. From these we obtain easy proofs of (BW theorem): Every bounded sequence in a Euclidean space has a convergent subsequence, and every Euclidean space is complete.