So the statement i want to prove is the following:
$a > 0 \Leftrightarrow a+a^{-1} >1$ with $a\in R$
So far I've been able to prove that:
when $0<a<1$, then $a^{-1} >1$
and when $a>1$, then $a^{-1}<1$
But where do i go from here?
Can i just come to the conclusion that "Therefore $a+a^{-1}>1$ ?
If $a>0$ then $a^{-1}>0$
Now if $a$ is an integer, then $\frac1a$ is a real number between $0$ and $1$. So when we add them we get a real number which is greater than $a$. The smallest value $a$(if an integer) is $1$ so $a+a^{-1}>1$.
Now if $a$ is a real number between $0$ and $1$ then $\frac1a>1$ which alone proves the statement.