Monotone Sequences
Definition 2.3.1
A sequence \(\{a_n\}\) is said to be
(a) monotone increasing if $a_n \leq a_{n+1}$ for all $n\in\mathbb{N}$
(b) monotone decreasing if $a_n \geq a_{n+1}$ for all $n\in\mathbb{N}$
(c) monotone if it is either monotone increasing or monotone decreasing
Theorem 2.3.2 (Monotone Convergence Theorem)
If a sequence \(\{a_n\}\) is monotone and bounded, then it converges.
<proof> Suppose that \(\{a_n\}\) is bounded and monotone increasing. Put \(\begin{align} E:=\{a_n:n\in\mathbb{N}\} \neq \emptyset \end{align}\) Then $E$ is bounded above and nonempty set. By the least upper bound property, we have $\alpha=\sup E$. Now, we want to show that \(\begin{align} \lim_{n\to\infty}a_n = \alpha \end{align}\) Let $\epsilon >0$ be given. Then there exists $n_0\in\mathbb{N}$ s.t. $\alpha-\epsilon <a_{n_0}\leq \alpha$. Since \(\{a_n\}\) is increasing, $a_n\geq a_{n_0}$ for all $n\geq n_0$. Thus, if $n\geq n_0$, then \(\begin{align} \alpha-\epsilon < a_n \leq \alpha<\alpha+\epsilon \end{align}\) $\therefore \displaystyle{\lim_{n\to\infty}a_n=\alpha}$ \(\tag*{$\square$}\)
Corollary 2.3.3 (Nested interval property)
Let \(\{I_n\}\) be a sequence of closed and bounded intervals with $I_n\supset I_{n+1}$ for all $n\in\mathbb{N}$.
$\Rightarrow \cap_{n=1}^\infty I_n\neq\emptyset$
<proof> Let $I_n = [a_n, b_n]$. Since $I_n \supset I_{n+m}$ for all $m\geq 0$, \(\begin{align} a_n \leq a_{n+m} \leq b_{n+m} \leq b_m \end{align}\) Thus, $a_n \leq b_m$ for all $m\in\mathbb{N}$. Since \(\{a_n\}\) is monotone increasing, by the monotone convergence theorem, we know that \(\{a_n\}\) converges to \(a=\sup\{a_n: n\in\mathbb{N}\}\). Now, note that $a\leq b_m$ for all $m\in\mathbb{N}$. So, we have $a\in I_m$ for all $m\in\mathbb{N}$.
$\therefore a\in \bigcap_{n=1}^\infty I_m$ \(\tag*{$\square$}\)
Examples 2.3.5
$e:= \displaystyle{\lim_{n\to\infty}}(1+\frac{1}{n})^n$
<proof>
Define $t_n:= (1+\frac{1}{n})^n$. Then,
\(\begin{align}
\begin{split}
(1+\frac{1}{n})^n&=1 + {n \choose 1}\frac{1}{n} + {n \choose 2}\frac{1}{n^2} + \cdots {n \choose n-1}\frac{1}{n^{n-1}} + {n \choose n}\frac{1}{n^n} \\
&= 1 + 1 + \frac{n(n-1)}{2}\frac{1}{n^2}+\cdots + \frac{n(n-1)\cdots 2\cdot1}{1\cdot2\cdots n}\frac{1}{n^n}
\end{split}
\end{align}\)
$(k+1)^{th}$ term of $t_n$ is
\(\begin{align}
\frac{n(n-1)\cdots (n-k+1)}{1\cdot 2\cdots k}\frac{1}{n^k}=\frac{1}{k!}(1-\frac{1}{n})\cdots (1-\frac{k-1}{n})
\end{align}\)
Then $t_{n+1}$ is
\(\begin{align}
(1+\frac{1}{n+1})^{n+1} =1 + 1 + \frac{(n+1)n}{2}\frac{1}{(n+1)^2} + \cdots \frac{(n+1)n\cdots 1}{1\cdot 2\cdots (n+1)}\frac{1}{(n+1)^{n+1}}
\end{align}\)
$(k+1)^{th}$ term of $t_{n+1}$ is
\(\begin{align}
\frac{1}{k!}(1-\frac{1}{n+1})\cdots (1-\frac{k-1}{n+1})
\end{align}\)
Since $(k+1)^{th}$ of $t_n$ is less than that of $t_{n+1}$ for all $k$, we have $t_n<t_{n+1}$ for all $n\in\mathbb{N}$. Moreover,
\(\begin{align}
\begin{split}
(1+\frac{1}{n})^n &=1 + 1 + \frac{n(n-1)}{2}\frac{1}{n^2}+\cdots + \frac{n(n-1)\cdots 2\cdot1}{1\cdot2\cdots n}\frac{1}{n^n}\\
&<1+1 + \frac{1}{2!} + \cdots + \frac{1}{n!} \\
&\leq 1 = 1 +\frac{1}{2} +\frac{1}{2^2} + \cdots + \frac{1}{2^{n-1}} \\
&=1+\frac{1-(\frac{1}{2})^n}{1-\frac{1}{2}} \\
&\leq 3
\end{split}
\end{align}\)
By the monotone convergence theorem, \(\{t_n\}\) converges.
\(\tag*{$\square$}\)
Definition 2.3.6
\(\begin{align} \lim_{n\to\infty}a_n =\infty &\iff \forall M>0, \exists N\in\mathbb{N} \text{ s.t. } n\geq N \Rightarrow a_n > M \\ \lim_{n\to\infty}a_n =-\infty &\iff \forall M<0, \exists N\in\mathbb{N} \text{ s.t. } n\geq N \Rightarrow a_n < M \\ \end{align}\)
Definition 2.4.1
Let \(\{p_n\}\) be a sequence and let \(\{n_k\}\) be strictly increasing sequence, i.e. $n_1 < n_2 <n_3 < \cdots$. We call \(\{p_{n_k}\}_{k=1}^\infty\) a subsequence of \(\{p_n\}\).
Theorem 2.4.3
Let \(\{p_n\}\) be a convergent sequence with $\displaystyle{\lim_{n\to\infty}p_n=p}$. Then for all subsequences of \(\{p_n\}\) converges to $p$.
<proof>
Let $\epsilon >0$ be given and let \(\{p_{n_k}\}\) be a subsequence of \(\{p_n\}\). Since $p_n\to p$ as $n\to\infty$, there exists $N\in\mathbb{N}$ such that \(\begin{align}
n\geq N \Rightarrow |p_n-p| <\epsilon
\end{align}\)
Now, $n_k \geq k$. So, if $k\geq N$, then $n_k\geq k \geq N$.
$\therefore |p_{n_k}-p|<\epsilon$
$\therefore \displaystyle{\lim_{k\to\infty}}p_{n_k}=p$
\(\tag*{$\square$}\)
Reference
- Manfred Stoll, 『Introduction to Real Analysis』, Pearson