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In my calculus course, there's example stated on the book:

Given that $M$ is an ordered set and the sequence $\{a_n\}\subset M$, prove that there's a (weakly) monotonic subsequence of $\{a_n\}$.

After some searching work, I realized that it's a special case (Hint: the color of $(j,k)$ depends on whether $a_j\le a_k$) of infinite Ramsey's theorem, well, and to my surprise, the finite one is implied in the infinite one. I read these two proofs. They are actually concise and graceful.

However, I cannot appreciate the aspect or essence of these two proofs. Observing more closely, I find that these proofs of infinite Ramsey's theorem use both constructive technique and non-constructive technique. For example, pigeonhole principle is non-constructive:

If $A\cup B$ is infinite, then $A$ is infinite or $B$ is infinite.

It seems that we cannot construct the infinite subset of $A$ or $B$ explicitly by the infiniteness of $A\cup B$. In the proof of infinite Ramsey's theorem, pigeonhole principle is used, but it's not the only technique. We should mention that sequence $\{a_n\}$ is constructed explicitly.

Now my question arises: How can we observe the infinity? How can we conceive these proofs on our owns? Maybe it's more a question about methodology and how to solve problems.

Thanks!

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Look deeply into the night sky. –  user02138 Oct 26 '12 at 16:10
    
The term "monotonous" is probably correct but I think what you mean is "monotone." –  daniel Oct 26 '12 at 16:31
    
@daniel Thanks. Edited. –  Frank Science Oct 26 '12 at 16:35
    
Based on the content o fthe question I felt like adding the tag (constructiv-mathematics) –  Hagen von Eitzen Oct 26 '12 at 16:36
    
You should note that why the pigeonhole principle is not constructive, it follows from a very basic understanding that the union of two finite sets is finite. Therefore in splitting an infinite set into two, at least one of the parts has to be infinite. –  Asaf Karagila Oct 26 '12 at 16:39
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