Exercise 2.5.8

Question

Another way to prove the Bolzano-Weierstrass Theorem is to show that every sequence contains a monotone subsequence. A useful device in this endeavor is the notion of a peak term. Given a sequence $\left(x_{n}\right)$, a particular term $x_{m}$ is a peak term if no later term in the sequence exceeds it; i.e., if $x_{m} \geq x_{n}$ for all $n \geq m$.
  \begin{enumerate}[label=(\alph*)] 
  \item Find examples of sequences with zero, one, and two peak terms. Find an example of a sequence with infinitely many peak terms that is not monotone.
  \item Show that every sequence contains a monotone subsequence and explain how this furnishes a new proof of the Bolzano-Weierstrass Theorem.
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
  \item $(1,2,\dots)$ has zero peak terms, $(1, 0, 1/2, 2/3, 3/4, \dots)$ has a single peak term, $(2, 1, 1/2, 2/3, \dots)$ has two peak terms (a similar argument works for $k$ peak terms) and $(1,1/2,1/3,\dots)$ has infinitely many peak terms.
    The sequence $(1, -1/2, 1/3, -1/4, \dots)$ has infinitely many peak terms, but is not monotone.
  \item Note that the (possibly finite) sequence of peak terms is monotonic decreasing. There are two possibilities - either there are infinitely many peak terms, or only finitely many. If there are infinitely many peak terms, simply take the subsequence of peak terms; if the parent sequence is bounded we have found a subsequence which converges (by the Monotone Convergence Theorem), hence proving BW in this case.

If there are finitely many peak terms, let the last peak term be at position $k$. Consider the terms after the last peak term. Since after this point there are no more peak terms, then for every term $x_n$ there must be at least one term $x_m \geq x_n$ where $m > n > k$. Therefore we can define the monotone subsequence $x'$ as $x'_1 = x_{k + 1}$, $x'_n$ as the first term after $x'_{n - 1}$ such that $x'_n > x'_{n-1}$. By MCT this subsequence converges, hence proving BW in this case as well.
   \end{enumerate}

Exercise 2.5.8

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Exercise 2.5.8

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