Very specyfic linear order not isomorphic with $\langle\mathbb N,\le\rangle$ Give an example of a numerable linear order such that each element has a successor, there is the smallest element, each element except the smallest one has a predecessor, but the order is not isomorphic with $\langle\mathbb N,\le\rangle$. I understand it all, but I can't come up with anything.

Consider $\Bbb N$ followed by $\Bbb Z$, i.e., $\\{0\\}\times \Bbb N\cup \\{1\\}\times \Bbb Z$ with lexicographic order. Or if you prefer it embedded into the real line: $\\{\,\arctan n\mid n\in\Bbb N\,\\}\cup\\{4+\arctan k\mid k\in\Bbb Z\,\\}$.