NoteNote. We will extend our definition of lim sup and lim inf to the unbounded sequences.Let $\{ a_n \}$ be a bounded sequence. Then $$A_n = \sup \{ a_n, a_{n+1}, ... \}$$ exists for every positive integers $n$. Since $\{ a_{n+1}, a_{n+2}, ... \} \subset \{ a_n, a_{n+1}, ... \}$, $A_n \leq A_{n+1}$ for all positive integers $n$, which means that $\{ A_n \}$ is decreasing. Since $\{ a_n \}$ is b..

Definition 20.1Definition 20.1. Let \(\{a_n\}\) be a bounded real sequence and let \(\mathcal{L}_a\) denote the set of all \(L\) such that \[ L = \lim_{k \to \infty} a_{n_k} \] where \(\{a_{n_k}\}\) is a convergent subsequence of \(\{a_n\}\). We define \[ \limsup_{n \to \infty} a_n = \sup \mathcal{L}_a \] and \[ \liminf_{n \to \infty} a_n = \inf \mathcal{L}_a \] The notations \(\overline{\lim}_{..

Theorem 19.1Theorem 19.1. Let \(\{a_n\}\) be a convergent sequence. Then for every \(\varepsilon > 0\), there exists a positive integer \(N\) such that if \(m, n \geq N\), then \[ |a_m - a_n| Proof. Let $\{ a_n \}$ be a sequence with the limit $L$ and let $\varepsilon > 0$. Then $\exists N \in \mathbb{P}$ such that $|a_n - L| Definition 19.2Definition 19.2. If \(\{a_n\}\) is a sequence such that..

The Bolzano-Weierstrass TheoremTheorem 18.1 (The Bolzano-Weierstrass Theorem). Every bounded real sequence has a convergent subsequence. Proof. Let $\{ a_n \}$ be a bounded real sequence. Then there exists a closed interval $[ c, d ]$ such that $a_n \in [ c, d ], \forall n \in \mathbb{P}$. Consider the two subinterval, $\left[ c, \frac{c+d}{2} \right], \left[ \frac{c+d}{2}, d \right]$. One of th..

Theorem 17.1 Theorem 17.1. If $x$ is a real number, there exists an increasing rational sequence $\{ r_n \}$ with limit $x$. Proof. By Theorem 7.8, $\exists r_1 \in \mathbb{Q}$ such that $x-1 Note that $$x - \frac{1}{n} RemarkRemark. If $a \geq 1$ and $x$ is a real number, we choose an increasing rational sequence $\{ r_n \}$ such that $\lim_{n \to \infty} r_n = x$. Since $r_n \leq r_{n+1}$, $a^..

Monotone SequencesDefinition 16.1. Let $\{a_n\}$ be a sequence. We say that $\{a_n\}$ is increasing (decreasing) if $a_n \leq a_{n+1}$ ($a_n \geq a_{n+1}$) for every positive integer $n$. We say that the sequence $\{a_n\}$ is monotone if either $\{a_n\}$ is increasing or $\{a_n\}$ is decreasing. If $a_n a_{n+1}$) for every positive integer $n$, we say that $\{a_n\}$ is strictly increasing (stri..

Divergent SequencesDefinition 15.1. Let $\{a_n\}$ be a sequence. We say that $\{a_n\}$ diverges to infinity (or minus infinity) and write \[ \lim_{n \to \infty} a_n = \infty \quad (\lim_{n \to \infty} a_n = -\infty) \] if for every real number $M$, there exists a positive integer $N$ such that if $n \geq N$, then $a_n > M$ ($a_n Theorem 15.2Theorem 15.2. Let $\{a_n\}$ and $\{b_n\}$ be sequences ..

Bounded Sequences Definition 13.1. We say that a sequence $\{a_n\}$ is bounded above (below) if there exists a number $M$ such that $a_n ≤ M$ ($a_n \geq M$) for every positive integer $n$. We say that $\{a_n\}$ is bounded if $\{a_n\}$ is bounded both above and below. Remark. A sequence $\{a_n\}$ is bounded $\iff$ $\exists M > 0$ such that $|a_n| \leq M, \forall n \in \mathbb{P}$. Theorem 13.2 T..

Subsequences Definition 11.1. Let $\{a_n\}$ be a sequence. Let $f$ be a strictly increasing function from $\mathbb{P}$ into $\mathbb{P}$. The sequence $a_{f(n)}$ is called a subsequence of the sequence $\{a_n\}$. [The function $f$ is strictly increasing if $f(m) LemmaLemma. Let $f$ be a strictly increasing function from $\mathbb{P}$ into $\mathbb{P}$. Then $n \leq f(n), \forall n \in \mathbb{P}$..

Sequence Definition 10.1. Let $X$ be a set. A sequence of elements of $X$ is a function from $\mathbb{P}$ into $X$. 특별히 real sequence라고 하면 $X = \mathbb{R}$인 경우로, $\{ a_n \}^{\infty}_{n = 1}$ 혹은 $\{ a_n \}$으로 쓴다. $a_n$을 단독으로 쓰면 $n$th term이라고 부른다. Limit of a SequenceDefinition 10.2. Let $\{a_n\}$ be a sequence of real numbers. We say that has $\{a_n\}$ has limit $L \in \mathbb{R}$ if for every $\v..