How can I prove that $2^n\ge n^4$? I'm trying to prove the following inequality.
$$2^n\ge n^4 (n&gt;n_0)$$
I noticed that from $n = 20$, the following inequality holds, and I want to use induction for the proof. However I simply cannot find any way to prove whenever $n$ holds, $n+1$ also holds as well. Any help would be appreciated. Thanks!
You can use induction to show that $2^n \geq n^4$ for all $n \geq 16$. When $n = 16$, we have $2^{16} = 16^4$. Suppose that $2^n \geq n^4$ for $n \geq 16$. We will show that $2^{n+1} \geq (n+1)^4$:
$$
(n+1)^4 = \left(1 + \frac{1}{n}\right)^4 \cdot n^4 \leq \left(1 + \frac{1}{16}\right)^4 \cdot 2^n &lt; 2 \cdot 2^n = 2^{n+1}.
$$
Because $$2^n=(1+1)^n&gt;\binom{n}{9}&gt;n^4$$ for all $n\geq20$.

Also, $$2^n&gt;\binom{n}{5}&gt;n^4$$ for all $n\geq130$.