There is a well-known fact in infinite combinatorics asserting that for each infinite linear order $P$ there is a countable subset $R\subseteq P$ of order type either $\omega$ or $\omega^{\*}$ 


(by $\omega^{\*}$ I mean set of natural number with reversed order). It seems to be a non-trivial result - for example, one can derive it from the Baumgartner-Hajnal theorem but this is, in my taste, too heavy machinery. 


Do you know who iss responsible for this result? Are there any cheaper ways (than the BH-theorem) to obtain it? 



Extracting countable chains from linear orders
It seems to me that a natural solution is to use Ramsey's theorem $\aleph\_0 \to (\aleph\_0)^2\_2$: enumerate a countable subset, and color two points depending on whether the enumeration agrees with the given order. 


This proof seems "cheaper" to me: Wlog the linear order $P$ is a subset of the rationals. Find a limit point $r$. Wlog $r$ is a limit point of the points from $P$ below $r$ -- so you can find an increasing sequence converging to $r$.


