Prove the infimum is the global positive minimum of f

Question

I just started reading a book on mathematical analysis and I'm having trouble understanding what exactly I need to prove and where to start (don't have much previous experience).

For $n \in \mathbb{R}$ with $n > 0$, let $f(n) = \frac{1}{n^2} + \frac{1}{n} + n^2$ and $i = inf\{f(n) : n > 0\}$. Prove that $i$ is a global positive minimum of $f$.

Isn't $i$ just the infimum of the function or is it something else? How could it be that a minimum could be lower? Thank you for any help!

Answer 1

I guess if you let $n$ be negative, then you find another minimum of $f$ smaller than $i$ but still positive. Here is the link to wolfram:

https://www.wolframalpha.com/input/?i=1%2Fx%5E2+%2B+1%2Fx+%2B+x%5E2