It would have been clearer if you had posted the whole argument. This is part of a proof by induction of what? Actually, it looks like the result of induction.

You know that for every n, so is certainly true for k= 1. Now suppose is true for some specific k andall n. Then a_n- (k+1)(L+ \epsilon)= [a_n- (L+ \epsilon)]- k(L+\epsilon)< a_{n+1}-k(L+\epsilon)[/tex] and now use [tex]a_n- k(L+ \epsilon)< a_{n+1}[/itex] with n+1 nstead of n- which you can do because it is true for all n.