1. Prove that grows more slowly than for any positive number .
Proof. So for , we have . If we fix we could probably prove this by induction on (e.g. fix ). Now can we somehow "take inverses" to deduce that for any positive number ?
2. Prove that for any , we have for all sufficiently large .
Proof. Let and . Since , has exponential growth. And has polynomial growth which is slower than exponential growth. Are these somewhat correct?