<s>[INST] Can someone please check if my reasoning for this proof is valid or not? I have already seen the other questions about this proof. I'm just trying a different sort of method, though I'm not sure if it's valid or not.
Context for the main question: Prove by induction $2^n\gt n^3$ for $n\ge10$
Obviously, the base case works for n=10
$1024=2^{10}\gt1000=10^3$
The induction hypothesis: Assume $P_n$ is true $\rightarrow$ $2^n\gt n^3$
I want to then prove that $2^{n+1}\gt (n+1)^3$
Now, using the induction hypothesis:
$2^n\gt n^3$
multiply both sides by 2
$2^{n+1}\gt 2n^3$
Using the fact that $n\ge10$ this implies that $n^3\ge10n^2$
$2n^3=n^3 +n^3\gt n^3 +10n^2=n^3 +3n^2 +7n^2$
Using the fact that $7\gt 1$ this implies that $7n\gt n$ since n is positive.
$n^3 +3n^2 +7n^2\gt n^3 +3n^2 +n^2$
Once again, using $n\ge 10$ this implies $n^2\ge 10n$
$n^3 +3n^2 +n^2\gt n^3 +3n^2 +10n=n^3 +3n^2 +3n+7n$
Again $7\gt 1$
$n^3 +3n^2 +3n+7n\gt n^3 +3n^2 +3n +n$
Using $n\ge 10$ one last time
$n^3 +3n^2 +3n +n\gt n^3 +3n^2 +3n+10$
Since $10\gt 1$
$n^3 +3n^2 +3n+10\gt n^3 +3n^2 +3n+1=(n+1)^3$
Thus, through the chain of inequalities, I have proved that $2^{n+1}\gt (n+1)^3$.
QED
Sorry if there are any errors in my reasoning. Thank you for reading and feedback.
 [/INST] It is ok, but I think you can go faster.
If you want to prove that:
$$2n^3>(n+1)^3\Leftrightarrow 2>\left(\frac{n+1}{n}\right)^3=\left(1+\frac{1}{n}\right)^3$$
You can do:
$$n\ge10\to\frac{1}{n}\le0.1\to \left(1+\frac{1}{n}\right)^3\le (1.1)^3=1.331<2$$
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