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Let $\left\{a_{n}\right\}$ be a sequence of real numbers such that $|a_{n+1}-a_{n}|\leq \frac{n^{2}}{2^{n}}$ for all $n \in \mathbb{N}$. Then

  1. The sequence $\left\{a_{n}\right\}$ may be unbounded.
  2. The sequence $\left\{a_{n}\right\}$ is bounded but may not converge.
  3. The sequence $\left\{a_{n}\right\}$ has exactly two limit points.
  4. The sequence $\left\{a_{n}\right\}$ is convergent.
in Set Theory & Algebra by Boss (29.7k points) | 74 views

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