analysis

**Plato** · September 3rd 2009, 12:44 PM

Originally Posted by flower3

$\mbox {Let x_n be an unbounded sequence .prove that there is a subsequence x_{n_k }$
$\mbox {of x_n<br />such that \frac {1}{x_{n_k}} \to ^k 0$

We may as well assume the sequence is not bounded above.
$\left( {\exists x_{n_1 } } \right)\left[ {x_{n_1 } > \max \left\{ {1,x_1 } \right\}} \right]$ .
$\left( {\forall j > 1} \right)\left( {\exists x_{n_j } } \right)\left[ {x_{n_j } > \max \left\{ {j,x_{n_{j - 1} } } \right\}} \right]$ .

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