Definition:
- A sequence (an) set to be bounded above if there exists a real number k such that an<=k for all n belongs to N then k is called upper bound of the sequence (an).
- A sequence (an) set to be bounded below if there exists a real number such that an>=k for all n belongs to N then k is called lower bound of a sequence (an).
- A sequence (an) is set to be bounded sequence if it is bounded above and bounded below .
- A sequence (an) set to be bounded if there is a real number k such that mod an<=k for all n belongs to N.
Examples:
i) Consider the sequences 1,1/2,1/3,....1/n,.... Here, 1 is the least upper bound and 0 is the Greatest lower bound. Therefore, it is a bounded sequence.
ii) The sequence 1,2,3,...,n,..... bounded below is 1, but not bounded above, Greatest lower bound is 1.
iii) Any constant sequences is a bounded sequences. Here, least upper bound=Greatest lower bound,
the constant term of the sequence.
iv) 1,-1,1,-1,.... is a bounded sequence, upper bound is 1, lower bound is -1, least upper bound is 1, greatest lower bound is -1.
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