Math 480 Real Analysis Midterm Study Guides

Vocabulary flashcards covering fundamental definitions, properties, and named theorems across Math 480 Midterm 1 and Midterm 2 study guides.

Subset ($A \subseteq B$)

A relation between sets meaning every $x \in A$ is also in $B$.

Set Equality ($A = B$)

The condition where $A \subseteq B$ and $B \subseteq A$; to prove two sets are equal, both inclusions must be proved.

Set Difference ($A \setminus B$)

The set operation defined as $\{x \in A : x \notin B\}$.

Complement ($A^c$)

Everything in the ambient set not in $A$, satisfying De Morgan's laws: $(A \cup B)^c = A^c \cap B^c$ and $(A \cap B)^c = A^c \cup B^c$.

Upper Bound

A real number $M$ is an upper bound of a set $A$ if $a \le M$ for every $a \in A$.

Bounded Set

A set $A$ that has both an upper bound and a lower bound.

Supremum (Least Upper Bound)

A number $s = \sup(A)$ satisfying: (1) $s$ is an upper bound of $A$, and (2) $s \le M$ for every upper bound $M$ of $A$.

$\varepsilon$-Characterization of Supremum

An upper bound $s$ is $\sup(A)$ if and only if for every $\varepsilon > 0$, there exists $x \in A$ such that $s - \varepsilon < x$.

Infimum (Greatest Lower Bound)

The greatest lower bound of a set; the mirror image of the supremum.

Axiom of Completeness

The foundational property of the real numbers stating that every non-empty set of reals that is bounded above has a supremum in $\mathbb{R}$.

Archimedean Property

The properties stating that: (1) for every $x \in \mathbb{R}$, there exists $n \in \mathbb{N}$ with $n > x$; and (2) for every $x > 0$, there exists $n \in \mathbb{N}$ with $1/n < x$.

Dense Set

A set $X$ is dense in $\mathbb{R}$ if for every $a < b$, there exists $x \in X$ such that $a < x < b$.

Sequence

A function $x : \mathbb{N} \to \mathbb{R}$, written $(x_n)$.

Convergent Sequence ($x_n \to L$)

A sequence where for every $\varepsilon > 0$, there exists $N$ such that $|x_n - L| < \varepsilon$ for all $n > N$; equivalently, all but finitely many $x_n$ lie in the neighborhood $V_\varepsilon(L) = (L - \varepsilon, L + \varepsilon)$.

Negation of Sequence Convergence ($x_n \text{ does not } \to L$)

The logical statement: there exists $\varepsilon > 0$ such that for every $N$, there is some $n > N$ with $|x_n - L| \ge \varepsilon$.

Divergence to infinity

A sequence behavior where for every $M > 0$, there exists $N$ such that $x_n > M$ for all $n > N$.

Bounded Sequence

A sequence for which there exists $M > 0$ such that $|x_n| \le M$ for all $n$.

Monotone Increasing Sequence

A sequence where $m > n$ implies $x_m \ge x_n$.

Subsequence

A sequence $(x_{n_k})$ formed from $(x_n)$ given strictly increasing natural indices $n_1 < n_2 < n_3 < \dots$.

Monotone Convergence Theorem

The theorem stating that if a sequence $(a_n)$ is monotone increasing and bounded, it converges, and $\lim(a_n) = \sup(\{a_n : n \in \mathbb{N}\})$.

Sandwich (Squeeze) Theorem

The theorem stating that if $a_n \le b_n \le c_n$ for all $n$, and $a_n \to L$ and $c_n \to L$, then $b_n \to L$.

Cauchy Sequence

A sequence where for every $\varepsilon > 0$, there exists $N$ such that $|a_n - a_m| < \varepsilon$ for all $n, m > N$.

Bolzano–Weierstrass Theorem

The theorem stating that every bounded sequence has a convergent subsequence.

Partial Sums ($s_n$)

The terms defined by the finite sum $s_n = x_1 + x_2 + \dots + x_n$.

Convergent Series

An infinite series $\sum x_n$ whose sequence of partial sums converges, with value $\sum x_n = \lim(s_n)$.

Cauchy Criterion for Series

$\sum a_n$ converges if and only if for every $\varepsilon > 0$, there exists $N$ such that $|a_{m+1} + \dots + a_n| < \varepsilon$ for all $n > m > N$.

Absolutely Convergent Series

A series $\sum a_n$ such that the series of absolute values $\sum |a_n|$ converges.

Conditionally Convergent Series

A series $\sum a_n$ that converges, but whose absolute series $\sum |a_n|$ diverges.

Geometric Series

A series of the form $\sum a r^k$, which converges to $\frac{a}{1 - r}$ when $|r| < 1$.

Injective Function

A function $f$ where $f(x) = f(y) \Rightarrow x = y$ (nothing is counted twice).

Surjective Function

A function $f$ where for every $y$, there exists $x$ such that $f(x) = y$ (everything is counted).

Bijective Function

A function that is both injective and surjective.

Same Cardinality ($|A| = |B|$)

The relationship meaning a bijection $A \to B$ exists between sets $A$ and $B$.

Countable Set

A set $X$ for which a bijection $\mathbb{N} \to X$ exists.

Uncountable Set

An infinite set that is not countable, such as $\mathbb{R}$ or $[0, 1]$.

Accumulation Point

A point $c$ of a set $A$ such that every interval $(c - \delta, c + \delta)$ contains a point of $A$ other than $c$.

Functional Limit ($\varepsilon$-$\delta$ Formulation)

$\lim_{x \to c} f(x) = L$ if for every $\varepsilon > 0$, there exists $\delta > 0$ such that $0 < |x - c| < \delta \Rightarrow |f(x) - L| < \varepsilon$.

Functional Limit (Sequential Formulation)

$\lim_{x \to c} f(x) = L$ if for every sequence $x_n \to c$ with $x_n \neq c$, the image sequence satisfies $f(x_n) \to L$.

Continuous Function at a Point

A function $f$ satisfying $\lim_{x \to c} f(x) = f(c)$.

Uniformly Continuous Function

A function $f$ on $A$ such that for every $\varepsilon > 0$, there exists $\delta > 0$ where $|x - y| < \delta \Rightarrow |f(x) - f(y)| < \varepsilon$ for all $x, y \in A$.

Open Set

A subset $A \subseteq \mathbb{R}$ where for every $x \in A$, there exists $\varepsilon > 0$ such that $(x - \varepsilon, x + \varepsilon) \subseteq A$.

Closed Set

A subset $C \subseteq \mathbb{R}$ where every Cauchy sequence in $C$ has its limit in $C$.

Compact Set

A set $A$ where every sequence has a subsequence converging to a point of $A$; in $\mathbb{R}$, this is equivalent to being closed and bounded.

Relatively Open Set in $A$

A set $B \subseteq A$ that can be written as $B = A \cap U$ for some open set $U$.

Connected Set

A set that cannot be written as $B_1 \cup B_2$ with $B_1, B_2$ disjoint, relatively open, and both non-empty; in $\mathbb{R}$, connected sets are intervals.

Extreme Value Theorem (EVT)

The theorem stating that if $f$ is continuous on a compact set $A$, then $f$ attains its maximum and minimum on $A$.

Intermediate Value Theorem (IVT)

The theorem stating that if $f$ is continuous on $[a, b]$ and $L$ is strictly between $f(a)$ and $f(b)$, then there exists $c \in (a, b)$ such that $f(c) = L$.

Little-o Notation

The asymptotic notation $f(x) = o(g(x))$ as $x \to x_0$ meaning $\lim_{x \to x_0} \frac{f(x)}{g(x)} = 0$.

Derivative (Quotient Form)

The limit $f'(x_0) = \lim_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0}$, provided this limit exists.

Derivative (Best Linear Approximation)

The characterization of differentiability at $x_0$ given by $f(x) = f(x_0) + f'(x_0)(x - x_0) + o(x - x_0)$.

Fermat's Theorem (Interior Extremum)

The theorem stating that if $f$ is differentiable on $(a, b)$ and attains an extremum at an interior point $c \in (a, b)$, then $f'(c) = 0$.

Darboux's Theorem

The theorem stating that if $f$ is differentiable on an open interval containing $[a, b]$ and $f'(a) < \alpha < f'(b)$, then there exists $c \in (a, b)$ such that $f'(c) = \alpha$.