DislikedFor example, suppose you chose your reals to be high of bars. The on the daily chart if you set n=2 then you need to look in the DNA for 5 bars. Let us suppose the highs of the last 4 bars are 1.2344, 1.2566, 1.2211, 1.2222, ????? What will be ????? Since there must be a decreasing or increasing 3-subsequence, the high of the next bar must be above 1.2222 or below 1.2211. Just two options.Ignored

not sure how you can conclude that the next bar must be above 1.2222 or below 1.2211. The next bar can be any number (thats really what the sub sequence theorem says and no matter what the numbers are, there will be sequence of n+1 that is either increasing or decreasing).

for e.g. next bar's high can be 1.2220 (between 1.2222 and 1.2211). that will still create a sequence of 2+1 bars

1.2344 1.2222 1.2220

I think you are thinking sub sequence has to be continuous. there is no such claim in the theorem. sub sequence will not always be continuous.

it is easy to create sequence of real numbers where no continuous sub sequence of more than 2 length exist no matter what n is

1 2 1.01 2.01 1.001 2.001 1.002 2.002 1.003 2.003 .......

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