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!