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#### IIT JAM MS 2020 Section A Problem 1 Solution

##### Problem

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

### Hints

##### Hint 1

If was bounded, show that , by sandwich theorem.

##### Hint 2

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

##### Hint 3

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.

#### Food For Thoughts

• What if, ?
• Find examples.

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