L15 Monotonic Sequence Theorem - and is bounded below then f a n g is convergent 1 ex f a n g is a monotonic increasing sequence such that a 1 = 2;a n 1

# L15 Monotonic Sequence Theorem - and is bounded below then...

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Lecture 15: Sequences (II) Monotonic Sequence Thmorem A sequence is increasing if a n < a n +1 ; decreasing if a n > a n +1 : A sequence is mono- tonic if it is either increasing or decreasing. A sequence is bounded above if there is a number M such that a n M for all n 1; bounded below if there is a number m such that m a n for all n 1. A sequence is bounded if it’s bounded above and below. Monotonic Sequence Thm: Every bounded, monotonic sequence is convergent. Alternative statements of the theorem: If a sequence f a n g is monotonically increasing and is bounded above, then f a n g is convergent. If a sequence f a n g is monotonically decreasing and is bounded below, then f a n g is convergent. 1
ex. f a n g is a monotonic increasing sequence such that a 1 = 2 ; a n +1 = 1 2 ( a n + 6) for n 2 : and a n < 6 for all n . Find the limit of the sequence. ex. f a n g is a monotonically decreasing sequence such that a 1 = 2 ; a n +1 = 1 3 a n for n 1 : and < a n 2 for all n . Find the limit of the sequence. 2
NYTI: Sequence p 2 ; p 2 p 2 ; q 2 p 2 p 2 ::: is mono- tonically increasing and bounded above. Find the limit of the sequence. 3