Can we prove that every bounded sequence has either an increasing or decreasing (or both) subsequence (without first proving the Bolzano-Weierstrass Theorem)?
2026-04-05 14:18:01.1775398681
Proving that every sequence has either an increasing or decreasing subsequence without Bolzano Weierstrass Theorem
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Yes, you can. You can refer to Theorem 3.4.7 on Page 80.
Given a sequence $(x_n)$, we can define $x_m$ is a peak if $x_m\ge x_n$ for all $n\ge m$. Then we separate the discussion into two cases, that is, $(x_n)$ has infinite many peaks (hence a decreasing subsequence) or $(x_n)$ has finite many peaks (hence an increasing subsequence).