# Thread: Prove every unbounded seq. contains a monotone subseq. that diverges to infinity

1. ## Prove every unbounded seq. contains a monotone subseq. that diverges to infinity

Could anyone explain to me how I would go about proving such a statement? Thank you and good day.

2. Originally Posted by binkypoo
Could anyone explain to me how I would go about proving such a statement? Thank you and good day.

Hint: let {x_n} be the unbounded seq. Choose x_n(1) to be s.t. x_n(1) > 1; now choose x_n(2) so that x_n(2) > max{ x_n(1), 2},....etc.
Can you see WHY can we always choose as above and how this solves the problem?

Tonio