The Limit of a Sequences

Sequence

Definition 1. A sequence is a function whose domain is $\mathbb{N}$.

고등학교에서는 수열을 '수의 나열'이라고 정의하곤 하는데, 정의에 의하면 꼭 '수'를 나열한 것만이 수열이 될 필요는 없다. 수가 아닌 함수나 다른 대상도 가능하다. 

Bounded Sequence

Definition 2. A sequence $\{ a_n \}$ is said to be bounded if its range is bounded. That is, there exists a number $M > 0$ such that $|a_n| \geq M$ for all $n \in \mathbb{N}$.

Monotonic Sequence

Definition 3. A sequence $\{ a_n \} \subset \mathbb{R}$ is said to be
(1) monotonically increasing if $a_n \leq a_{n+1}$ for all $n \in \mathbb{N}$;
(2) monotonically decreasing if $a_n \geq a_{n+1}$ for all $n \in \mathbb{N}$.
A sequence is said to be monotone if it is either monotonically increasing or decreasing.

The Monotonic Sequence Theorem

Theorem 1. Suppose $\{ a_n \}$ is monotonic. Then $\{ a_n \}$ converges if and only if it is bounded.

Proof. ($\Longrightarrow$) Let $\{ a_n \}$ converge to $L$. Take any $\epsilon > 0$. Then there is $N \in \mathbb{N}$ such that $\forall n \geq N \Longrightarrow |a_n - L| < \epsilon$. This means that if $n \geq N$, then $L - \epsilon < a_n < L + \epsilon$. Let $x = \max \{ a_1, ..., a_{N-1}, L+\epsilon \}$ and $y = \min \{ a_1, ..., a_{N-1}, L - \epsilon \}$. Then $\forall n \in \mathbb{N}$, $y \leq a_n \leq x$. Thus $\{a_n\}$ is bounded above.
($\Longleftarrow$) By the least upper bound property, there exists a supremum $L$ of $\{a_n \}$. Then $a_n \leq L, \forall n \in \mathbb{N}$. For any $\epsilon > 0$, there always exists $N \in \mathbb{N}$ such that if $n \geq N$, then $L - \epsilon < a_n \leq L < L + \epsilon$. Therefore $|a_n - L| < \epsilon, \forall n \leq N$. This means that $a_n$ converges to $L$. $\blacksquare$