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