I am trying to prove that $e^x>1+x^2$ for any $x>0$ for my homework assignment.
However I have run into trouble doing this. I was trying to probe that $\ln {{e}^{x}}>\ln (1+{{x}^{2}})$ is true for $x>0$ and then that would mean that $e^x>1+x^2$ is true because $\ln x$ is a monotone rising function.
However I have come to the following conclusion$\frac{{{x}^{2}}}{1+{{x}^{2}}}\le \ln (1+{{x}^{2}})\le {{x}^{2}}$
which means $x\le \frac{{{x}^{2}}}{1+{{x}^{2}}}$ must be true. but it is not.
I am wondering where I made a mistake here - Or perhaps where I made many mistakes?
Maybe there is a much better why to solve this question also?
Thanks a lot :)