Elementary Analysis: The Theory of Calculus Vocabulary

Vocabulary flashcards covering fundamental concepts of real analysis including sets of numbers, axioms, sequence and series convergence, continuity, and the theory of integration.

Natural Numbers ($\mathbb{N}$)

The set of all positive integers, denoted as ${1, 2, 3, \dots}$. Each element $n$ has a successor $n + 1$.

Peano Axioms

Five postulates providing the foundation for natural numbers, including the basis for mathematical induction.

Rational Numbers ($\mathbb{Q}$)

Numbers that can be expressed in the form $\frac{m}{n}$ where $m, n \in \mathbb{Z}$ and $n \neq 0$. They contain all terminating decimals.

Algebraic Number

A number that satisfies a polynomial equation $c_nx^n + \dots + c_1x + c_0 = 0$ where the coefficients are integers.

Ordered Field

A field equipped with an ordering relation $\leq$ that satisfies the transitive law, preserves addition ($a \leq b \implies a+c \leq b+c$), and preserves multiplication by non-negative values.

Completeness Axiom

The property stating that every non-empty subset of $\mathbb{R}$ that is bounded above has a least upper bound (supremum).

Archimedean Property

The principle stating that for any a > 0 and b > 0, there exists a positive integer $n$ such that na > b.

Convergent Sequence

A sequence $(s_n)$ that approaches a real number $s$ such that for every \epsilon > 0, there exists a number $N$ where n > N \implies |s_n - s| < \epsilon.

Monotone Sequence

A sequence that is either non-decreasing ($s_n \leq s_{n+1}$) or non-increasing ($s_n \geq s_{n+1}$) for all $n$.

Cauchy Sequence

A sequence where for every \epsilon > 0, there exists $N$ such that m, n > N \implies |s_n - s_m| < \epsilon. Constant for real numbers, this is equivalent to convergence.

Subsequence

A sequence $(t_k)$ formed by selecting terms of a sequence $(s_n)$ in their original order, defined as $t_k = s_{n_k}$ where n_1 < n_2 < n_3 < \dots.

Bolzano-Weierstrass Theorem

A fundamental theorem stating that every bounded sequence has a convergent subsequence.

Metric Space

A set $S$ equipped with a distance function $d$ (metric) satisfying non-negativity, symmetry, and the triangle inequality.

Compact Set

In $\mathbb{R}^k$, a set that is closed and bounded. Formally, a set where every open cover has a finite subcover.

Ratio Test

A series convergence test determined by calculating $\lim \sup |\frac{a_{n+1}}{a_n}|$.

Uniform Convergence

A type of convergence for a sequence of functions $(f_n)$ on a set $S$ where the value of $N$ depends only on $\epsilon$, not on $x \in S$. Formally, \forall \epsilon > 0, \exists N \text{ s.t. } n > N \implies |f_n(x) - f(x)| < \epsilon \text{ for all } x \in S.

Riemann Integral

The limit of Darboux sums (upper and lower) as the partition becomes finer, representing the area under a curve.

Fundamental Theorem of Calculus (Part I)

States that if $g$ is differentiable on $(a, b)$ and $g'$ is integrable, then $\int_a^b g'(x)dx = g(b) - g(a)$.

Taylor Series

A power series representation of a function $f$ about a point $c$, given by $\sum_{k=0}^{\infty} \frac{f^{(k)}(c)}{k!}(x-c)^k$.