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If is a sequence of real numbers such that , then

(A) is a bounded sequence

(B) is an unbounded sequence

(C) is a convergent sequence

(D) is a monotonically decreasing sequence

If was bounded, show that , by sandwich theorem.

If was convergent, show that , by algebra of limits.

If was motonotically decreasing and bounded below, then it would have been convergent by Monotone Convergence Theorem.

Let's consider if it is not below below, i.e.

- Take
- Take
- Take

Find the limit in each of this case.

Hence, it will be unbounded. See the full solution and proof idea below.

- What if, ?
- Find examples.

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