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..

IntegersDefinition 7.1. The set of integers, denoted $\mathbb{Z}$, is the set $$\{0\} \cup \mathbb{P} \cup -\mathbb{P}$$ where $- \mathbb{P} = \{ -n \, | \, n \in \mathbb{P} \}$. RationalsDefinition 7.2. The set of rational numbers, denoted $\mathbb{Q}$, is the set $$\left\{ \frac{p}{q} \;\middle|\; p, q \in \mathbb{Z} \text{ and } q \neq 0 \right\}$$Integer ExponentsDefinition 7.3. Let $x \in \..

Successor Set Definition 6.1. A set $X \subseteq \mathbb{R}$ is said to be a successor set (i) if $1 \in X$,(ii) if $n \in X$, then $n+1 \in X$.실수 집합 $\mathbb{R}$은 그 자체로 successor set이기 때문에,(A9, A1) 위와 같이 정의한 successor set은 적어도 하나는 존재한다고 말할 수 있다. Lemma 6.2Lemma 6.2. If $\mathcal{A}$ is any nonempty collection of successor sets, then $\cap \mathcal{A}$ is a successor set. Proof. Since $1$ is in e..

Binary OperationDefinition 3.1. A binary operation on a set $X$ is a function from $X \times X$ into $X$. Real Number Definition 3.2. The real numbers $\mathbb{R}$ is a set of objects satisfying Axioms 1 to 13 as listed in the following:(A1) There is a binary operation, called addition and denoted $+$, such that $x, y \in \mathbb{R} \Longrightarrow x + y \in \mathbb{R}$,(A2) $(x + y) + z = x + (..

주어진 행렬을 LU decomposition을 한 뒤 $L$과 $U$ 행렬 각각의 inverse를 구하면 역행렬을 빠르게 구할 수 있다. 이때 triangular matrix의 inverse를 빠르게 계산하는 방법을 소개하려고 한다. 예컨대 다음과 같은 상삼각 행렬의 역행렬을 계산해 보자. $$U = \begin{pmatrix} 3 & 6 & 8 \\ 0 & 4 & 7 \\ 0 & 0 & 5 \end{pmatrix}$$ 이때 각각의 성분에 대해서 따로따로 생각해 보자. Gauss elimination을 생각하면 역행렬의 대각 성분은 원행렬의 대각 성분의 역수가 된다. $$U^{-1} = \begin{pmatrix} \frac{1}{3} & * & * \\ 0 & \frac{1}{4} & * \\ 0 & ..

우리는 주어진 곡선 $C$가 얼마나 휘어있는지를 측정하고자 한다. 직관적으로 생각해보면 같은 원이라고 하더라도 반지름이 클수록 원은 국소적으로 덜 휘어있고, 반지름이 작을수록 더 휘어있다고 말할 수 있다. 그렇다면 이러한 곡선의 휜 정도를 나타내는 값, 즉 곡률을 어떤 곡선의 방정식이 주어졌을 때 어떤 방식으로 정의할 수 있을까? 한 가지 떠올릴 수 있는 방법은 tangent vector를 이용한 방법이다. 곡선의 tangent vector, 즉 속도는 곡선의 진행 방향과 평행한 접선 벡터인데, 곡선이 많이 휘어있다면 이동경로가 많이 뒤틀린다는 뜻이고 그만큼 tangent vector의 변화량이 크다는 뜻이다. 이는 곡률을 tangent vector의 변화량으로 측정할 수 있음을 시사한다. 이 값은 물리..

GroupDefinition 1. A group $\langle G, \ast \rangle$ is a set $G$, closed under a binary operation $\ast$, such that the following axioms are satisfied:(1) $\forall a, b, c \in G$, $(a \ast b) \ast c = a \ast (b \ast c)$(2) $\exists e \in G$ such that $\forall x \in G$, $e \ast x = x \ast e = x$(3) $\forall a \in G$, $\exists a' \in G$ such that $a \ast a' = a' \ast a = e$.