Why is this a flawed proof?

Knowing that $a$ is an element in $A$ and $b$ is an element in $B$. $R$ being a symmetric binary relation:

> 
> “Consider any $a$ and $b$ such that $aRb$.  
> 
>  Since $R$ is symmetric, we have $bRa$.  
> 
>  Since $aRb$ and $bRa$, we have $bR^{−1}a$ and $aR^{−1}b$.  
> 
>  Since our choice of $a$ and $b$ was arbitrary, we have proven that $R^{−1}$ is symmetric.”
> 
> 
> 

There's apparently a flaw with this, but I can't for the life of me figure it out.

[Homework Question]

Thanks in advance.