Fix any errors in my proof (beginner)

You can not proof statements just by giving examples. Your proof must be general and work for all possible cases. This is how I would prove it:

Assume 
\[(1)         a<1/a<b<1/b.\]

Since $a<\frac{1}{a}$, we have
\[a-\frac{1}{a}<0  \Rightarrow \frac{a^2-1}{a}<0.\]
There are two cases that the above inequality can hold:

Case 1. $a^2-1<0$ and $a>0$. Then we must have $0<a<1$. Thus 
\[1<\frac{1}{a}<b \Rightarrow  b>1.\]
This is a contradiction because if $b>1$, then $b>1/b$ which violates the assumption (1)  that $b<1/b$. Thus Case 1 is not possible. 

Case 2. $a^2-1>0$ and $a<0$. Then we must have $a<-1$. 

Therefore if the assumption (1) holds, then $a<-1$.