1) There is a natural number 1.
2) For every natural number K, there is a natural number K+1.
Conclusion: there is an infinite natural number.
The premises 1 and 2 are correct, but I think we both know there is no infinite natural number. So the conclusion can't be made.
Arbitrarily long: for any given natural number K, there is a monotone sub-sequence of length L>K. Proven.
Infinitely long: there is a monotone sub-sequence S such that for any given natural number K, S is more than K units long. Not proven.
bonanova, regarding countable vs uncountable infinity, my OP is unclear. I am only considering countable infinity.
Edited by Rainman, 09 September 2012 - 11:46 AM.