Give an example of a sequence that is bounded from above and bounded from below but is not convergent. One possibility is {(−1)n}+∞ n=1. = −1,1, ...

Example. Determine whether the sequence is strictly increasing or strictly decreasing. { }2. 1. n n n. ∞. −. = Page 15. Example. Determine whether the sequence.

Similarly, we define decreasing and strictly decreasing sequences. Sequences which are either increasing or decreasing are called monotone.

For example, the sequences (4), (5), and (7) are bounded above, while (6) is not. For (4) and (5), any number ≥ 1 is an upper bound. Theorem 1.3. A positive ...

Example. 4: The following sequences are not monotonic, but they are ultimately monotonic (i.e., monotonic after a certain number of terms): i) 5, 4, 7, 2, 13, ...

1. Any increasing and bounded sequence converges to its supre- mum. 2. Any decreasing and bounded sequence converges to its infi-.

20 апр. 2020 г. · The following theorem gives a very elegant criterion for a sequence to converge, and explains why monotonicity is so important. Monotone ...

Example. Let an = n. Then (an) is monotone increasing. So is an = (2n + 1)2 ... Let (an) be a bounded above monotone non-decreasing sequence. Then (an) ...

3 мар. 2018 г. · Theorem 3.4.4 (Monotone Convergence Theorem). Suppose hsni is a monotone sequence. Then hsni is convergent if and only if hsni is bounded.

Monotone sequences of events. Def.1.1: A sequence (An)n≥1 of events is increasing if An ⊂ An+1 for all n ≥ 1. It is decreasing if An ⊃ An+1 for all n ≥ 1.