Question

[Real analysis] Convergence of na^n, where a in [0,1)

Answer 1

This is just a hint. Try it and see if you get stuck.

The ratio between successive terms of the sequence tends to a. Therefore, if you pick some b with a < b < 1, the ratio is eventually <= b. That will allow you to compare the sequence with a geometric one of ratio b and apply the squeeze theorem, or some equivalent argument with epsilons.

Answer 2

Is there a reason you can't just use L'Hopital?

You could break the a^(n) into two roots. One to dominate the n, and one to go to 0.

If you need to prove exponentials beat polynomials, that seems like it involves calculus again, and you might as well go back to L'Hopital.