Prove that:
$\forall b \in \mathbb{R}, \forall c, n_0 \in \mathbb{N^+}, \exists n \in \mathbb{N^+} \text{ s.t. } n \geqslant n_0 \text{ AND } bn^2 < c2^n $.
I tried an Epsilon-Delta style proof and didn't get anywhere. Perhaps I did it wrong?
I have tried proof by contradiction and didn't was not sure what to set $n_0$ to. Anyone have any ideas on how to do this one?
It is enough to show that the sequence $(\frac{n^2}{2^n})$ is a null sequence. But this follows from $2^n=(1+1)^n=\sum_{j=0}^n \binom{n}{j}>\binom{n}3$ for $n\geq3$.