My thoughts: So apart from this step, the rest of the proof is fairly simple. Now in terms of lim sup, I know that it is just equal to the regular limit for convergent sequences and for bounded, divergent sequences it is what the "peaks" of the sequence converges towards. Of course there is a formal definition as well but I would like to have an intiutive understanding as well.
However I am not sure how to deduce the the inequality between the fraction and t from the definition of limsup... Can someone help me? Am I missing some key property of lim sup?