Give an example of the following or prove that such a request is impossible: (h) A sequence with an infinite number of ones (1's) that converges to a limit not equal to 1. (i) An unbounded sequence (an) and a convergent sequence (bn) with (an - bn) bounded.
Analysis
What is Sequence:
A sequence is a collection of objects where repeats are allowed and the order is important in mathematics. It has members, sometimes known as words or elements, just like a set. The length of the sequence depends on the number of elements. In contrast to a set, a sequence may include the same items more than once at various points, and unlike a set, the order of the sequence is important. A sequence can be described formally as a function from natural numbers (the elements of the sequence's locations) to the items that are present in each of those positions. A "indexed family" is a function that connects an index set, which may or may not be a set of integers, to another set of items.
To Give:
Two statements are Given:
A sequence with an infinite number of ones (1's) that converges to a limit not equal to 1.
An unbounded sequence and a convergent sequence with bounded.
Either we give an example or we disprove it.
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