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Sequences.pdf - ANALYSIS REAL VARIABLE SEQUENCES A...

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ANALYSIS REAL VARIABLE SEQUENCES A monotonic increasing sequence either converges to a finite limit or diverges to = . Proof Let S be the set of numbers in the sequence { a n } . Case (i) S has no upper bound. Given any X, n 0 / a n 0 > X a n > X for n n 0 . This means that the sequence diverges to + . Case (ii) S has an upper bound; bda n = a . Given ε > 0 n 1 / a n 1 > a - ε a - ε < a n a for n n 1 . This means that the sequence converges to a . Upper and lower limits Definition lim n →∞ sup a n lim n →∞ a n ) = lim n →∞ sup m n a m lim n →∞ inf a n lim n →∞ a n ) = lim n →∞ inf m n a m Justification (for lim sup) Case (i) Suppose S has no upper bound. i.e sup m 1 a m = + Then sup m n a m = + for all n . In these circumstances we say that lim n →∞ a n = + Case (ii) Now suppose S has an upper bound. Write sup m n a m = A ( n ) A (1) A (2) . . . is a monotonic decreasing sequence. If this sequence is bounded below, then it converges to a number Λ, and lim n →∞ a n = Λ, and if not, lim n →∞ a n = -∞
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