26. Convergent Subsequence

14 - Scenic Viewpoints Theorem

Scenic Viewpoints Theorem

Let $(a_{n})$ be a real sequence then
$(a_{n}) has a monotone subsequence$

Proof

Let $V = {k \in N : if m > k then a_{m} < a_{k}}$
I.e. the elements of $V$ are peaks
So if $k \in V$ then $a_{k}$ is higher than all terms after it

$V$ is infinite
Suppose the elements of $V$ are $k_{1} < k_{2} < \dots$
Then $(a_{k_{r}})_{r}$ is a subsequence of $(a_{n})$

Hence it is monotone decreasing as
$if r < s then k_{r} < k_{s} so a_{k_{r}} > a_{k_{s}}$

$V$ is finite
Then there is $N$ such that if $k \in V$ then $k < N$
Let $m_{1} = N$ . Then $m_{1} \neq = V$ so there is $m_{2} > m_{1}$ with $a_{m_{2}} \geq a_{m_{1}}$
As $m_{2} \neq \in V$ then there is $m_{3} > m_{2}$ with $a_{m_{3}} \geq a_{m_{2}}$
Repeating inductively, we can construct $m_{1} < m_{2} < m_{3} < \dots$ with

$a_{m_{1}} \leq a_{m_{2}} \leq a_{m_{3}} \leq \dots$
Then
$(a_{m_{r}})_{r} is an increasing subsequence of (a_{n})$

Link to original

15 - Bolzano-Weierstrass Theorem

Bolzano-Weierstass Theorem

Let $(a_{n})$ be a bounded real sequence then
$(a_{n}) has a convergent subsequence$

Proof

By the Scenic Viewpoints Theorem, $(a_{n})$ has a monotone subsequence
As the monotone subsequence is bounded (by assumption of whole)
Then by Monotone Subsequences Theorem, the subsequence converges

Link to original

Bolzano-Weierstrass Theorem for Complex Sequences corollary

Let $(z_{n})$ be a bounded complex sequence then
$(z_{n}) has a convergent subsequence$

Proof

Write $z_{n} = x_{n} + i y_{n}$ where $x_{n}, y_{n} \in R$
Suppose $(z_{n})$ is bounded by $M$ so tthat
$∣ z_{n} ∣ \leq M for all n$
Then $(x_{n})$ and $(y_{n})$ are also bounded by $M$ , and are real sequences
By Bolzano-Weierstrass Theorem then
$(x_{n}) has a convergent subsequence (x_{n_{r}})_{r}$
Now $(y_{n_{r}})_{r}$ is a bounded real sequence so by Bolzano-Weierstrass Theorem
$(y_{n_{r}})_{r} has a convergent subsequence (y_{n_{r_{s}}})_{s}$
As $(x_{n_{r_{s}}})_{s}$ is a subsequence of $(x_{n_{r}})_{r}$ then it converges
Hence by Convergence of Complex Sequences then $(z_{n_{r_{s}}})_{s}$ converges

Link to original

Oxford Maths Notes

Explorer

26. Convergent Subsequence

14 - Scenic Viewpoints Theorem

15 - Bolzano-Weierstrass Theorem

Graph View