Do the following. (a) Let {a,} be an decreasing sequence that is bounded below. State the monotone convergence theorem for this sequence.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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8. Do the following.
(a) Let {a„} be an decreasing sequence that is bounded below. State the
monotone convergence theorem for this sequence.
(b) Let {b,} be the sequence defined by b, = 4 + . Do the following
i. Show that {b,} is bounded below by 4.
ii. Show that b, bn+1 for all n e N.
iii. Explain why i. and ii. together imply that lim, bn exists, then find
this limit using limit properties.
iv. Let c be any real number with the property that c < b, for each n e N.
Use the Monotone Convergence Theorem, the definition of inf{b„}
and the result in i. to explain why c< 4.
Transcribed Image Text:8. Do the following. (a) Let {a„} be an decreasing sequence that is bounded below. State the monotone convergence theorem for this sequence. (b) Let {b,} be the sequence defined by b, = 4 + . Do the following i. Show that {b,} is bounded below by 4. ii. Show that b, bn+1 for all n e N. iii. Explain why i. and ii. together imply that lim, bn exists, then find this limit using limit properties. iv. Let c be any real number with the property that c < b, for each n e N. Use the Monotone Convergence Theorem, the definition of inf{b„} and the result in i. to explain why c< 4.
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