Rainman

To be more specific, you have proven that there are arbitrarily long monotone sub-sequences, but not that there are infinitely long ones.

I defined a sequence as a set, which means no two numbers can be equal. And a proof for each finite sequence is not quite enough to prove the infinite extension. For example, every finite sequence has a largest number but this doesn't imply that an infinite sequence has a largest number.

Show that every infinite sequence of real numbers contains an infinite monotone subsequence. --- Sequence: a set with a specified order of its elements. For example {1-2-4-3} is a different sequence from {4-1-3-2}. Subsequence: a subset retaining the order of the original sequence. For example {2-3-5-7} is a subsequence of {1-2-3-4-5-6-7}. Monotone: either increasing (each number is smaller than the next number) or decreasing (each number is larger than the next number).