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TIFR ECE 2023 | Question: 15

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Let $\left\{x_{n}\right\}_{n \geq 0}$ be a sequence of real numbers which satisfy

$$x_{n+1}\left(1+x_{n+1}\right) \leq x_{n}\left(1+x_{n}\right), \quad n \geq 0 .$$

Choose the correct option from the following.

  1. The sequence could be unbounded.
  2. The sequence is always bounded but does not necessarily converge.
  3. The sequence always converges to a non-zero limit.
  4. The sequence always converges to zero.
  5. None of the above.

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