We will show that every bounded sequence has a sequential cluster point. First recall what we mean when we say that c is a cluster point of the set S . The point c is a cluster point of the set S if and only if there is a sequence of elements of S , all different from c , and converging to c . We modify this lemma and present a result in terms of subsequences. Every bounded sequence has a sequential cluster point.
Every bounded sequence has a convergent subsequence.
L , then, given there is a natural number N such that, .We use this idea to define a Cauchy sequence. A convergent sequence is a Cauchy sequence. We answer this question bit by bit. Theorem 231 A Cauchy sequence of real numbers converges in . We now begin to use our knowledge of sequences and topology to understand some of the most basic results in Calculus.