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Unbounded sequence

A sequence of real number \{x_n\}_{n \in \mathbb{N}} satisfies the condition

    \[|x_n-x_m| > \frac{1}{n} \quad \text{whenever} \quad n<m\]

Prove that x_n is not bounded.

Solution

Let

    \[I_n= \left( x_n - \frac{1}{2n} , x_n + \frac{1}{2n} \right)\]

The inequality \left| x_n - x_m \right| > \frac{1}{n} for n<m implies that I_n \cap I_m = \varnothing for m \neq n. Further, let I = \bigcup \limits_{n=1}^{\infty} I_n. The sum of lengths of all the intervals I_n for n=1, 2, \dots is clearly infinite and hence I is unbounded and so is the sequence x_n.

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