Constructing an Increasing Sequence to Prove a Limit is Positive Infinity

[Icebreaker]

Suppose f:[a,b)->R is such that lim(x->b-)=+inf. Prove that there exists an increasing sequence {x_n} in (a,b) such that f(x_n)>n for all n.

I don't know where to start. It would be easy if I can prove that f is strictly increasing after some point. f might not be continuous so I can't simply look for all f(x) in the form x^2 or something... Too many possibilities. Any pointers will be helpful!

 

[benorin]

What do you have for a definition of lim(x->b-)=+inf ? This will tell you where to start.