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I'm trying to prove that a sequence of real numbers always includes a decreasing or increasing subsequence. At first I came up with the new sequence $y_n:=inf\{x_m|m \ge n\}$. I figured out $y_n$ is increasing and every member of $y_n$ is an element of R due to the GLBP. However I now realize that the members of $y_n$ need not be an element of $\{x_n\}$, since it's an infinum... I'm stuck here and would appreciate any help. Thanks.

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Let us call a natural number $n$ as suitable if $x_n > x_m$ for all $m > n$.

1) If the sequence contains infinitely many suitable points, you can check that these points form a decreasing subsequence.

2) If not, then after some large number $N_1$, all numbers are not suitable. If $N_1$ is not suitable, there is $x_{N_2} \geq x_{N_1}$ for some $N_2 > N_1$. Now, $N_2$ is also not suitable, so there is some $x_{N_3} \geq X_{N_2}$ for some $N_3 > N_2$.

Proceeding in this fashion, you will get a monotone increasing subsequence of the given sequence. Since one of the two cases must occur, the result follows.

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