I have to prove the following theorem:



I am not asking for the whole problem, but am stuck on the first part (Proving that output of c implies g(n) is in the order of f(n)).

I know the following:


  The order of f(n) is the set of complexity functions g(n) for which there exists some positive real constants c and d and some nonnegative integer N such that, for all n >= N:
  $c * f(n) &lt;= g(n) &lt;= d * f(n)$


At this point, I feel that I may be able to use induction with the formula above to show that the limit of g(n)/f(n) is nonnegative, but I can't quite wrap my head around that. Or perhaps there is another way to show that the limit converges?


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