In the book of Algebra by Hungerford, at page 15, it is asked that
However, doesn't what is asked to prove contradict with the statement that it follows ?
Moreover, consider $(\mathbb{R}, \leq)$, both $1$ and $2$ has no immediate successor, so isn't what is asked to prove wrong ?
The order on $\Bbb R$ is not a well-order since $\Bbb R$ itself is a subset without a minimal element (as is every open interval, by the way). So it is not a counterexample.
The example you are required to provide also does not contradict the statement that you need to prove. If anything, it just means that the linear order you're looking for is not a well-order.