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Why is the following statement on sequence convergence false?
I can't see why
"If 0 ≤ an ≤ bn for n ≥ 1 and {bn} converges, then {an} converges." is false.
1 Answer
- kbLv 710 years agoFavorite Answer
Let an = 2 + (-1)^n, while bn = 3 for n = 1, 2, ... .
That is, {an} = {1, 2, 1, 2, 1, 2, ...} and {bn} = {3, 3, 3, 3, 3, 3, ...}
Note that {bn} converges to 3, while {an} diverges because it has no unique limit.
(The even terms of an converge to 2, while the odd terms of an converge to 1.)
I hope this helps!
