TIFR ECE 2023 | Question: 15

Question

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.

• A. The sequence could be unbounded.
• B. The sequence is always bounded but does not necessarily converge.
• C. The sequence always converges to a non-zero limit.
• D. The sequence always converges to zero.
• E. None of the above.