Victor Kreiman
Home
Math 114 (Spring 26) Math 221 (Spring 26) Math 222 Math 301 Math 310 Math 350 (Spring 26) Math 451
Research

Math 350 | Advanced Calculus | Spring 2026

This study guide gives the main definitions and theorems covered in the course.

Contents

  1. The Real Numbers
  2. Cardinality
  3. Convergence of a Sequence
  4. Topology of \( \mathbb{R} \)
  5. Topology of Metric Spaces
The Real Numbers

Consider the following sets of numbers: \begin{align} \mathbb{N} &= \{1, 2, 3, \ldots \}\ \text{ (natural numbers)}\\ \mathbb{Z} &= \{\ldots,-3,-2,-1,0,1, 2, 3, \ldots \}\ \text{ (integers)}\\ \mathbb{Q} &= \left\{\frac{a}{b} : a,b\in \mathbb{Z}, b\neq 0 \right\}\ \text{ (rational numbers)}\\ \end{align} We have that \(\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q}\).

There does not exist \(p\in \mathbb{Q}\) such that \(p^2=2\).

In other words, \(\sqrt{2}\notin \mathbb{Q}\). We will shortly introduce the set of real numbers, \(\mathbb{R}\), a larger set of numbers than \(\mathbb{Q}\) which contains \(\sqrt{2}\). We will define \(\mathbb{R}\) rigorously as a set satisfying eleven axioms. The first ten of these axioms are given in the following three definitions.

A field is a set \(F\) together with two binary operations, denoted by \(+\) and \(\cdot\), which satisfy the following axioms:
  1. For all \(a,b\in F\), \(a+b = b+a\) and \(a\cdot b=b\cdot a\).
  2. For all \(a,b,c\in F\), \((a+b)+c = a+(b+c)\) and \((a\cdot b)\cdot c = a\cdot (b\cdot c)\).
  3. For all \(a,b,c\in F\), \(a\cdot (b + c) = a\cdot b + a\cdot c\).
  4. There exist elements \(0,1\in F\) for which \(0+a=0\) and \(1\cdot a = a\) for all \(a\in F\).
  5. For every \(a\in F\), there exists an element \(-a\in F\) for which \(a + (-a)=0\).
  6. For every \(a\in F\), \(a\neq 0\), there exists \(a^{-1}\in F\) for which \(a\cdot a^{-1}=1\).
Let \(S\) be any set. A total order (or linear order) on \(S\) is a relation \(\lt \) on \(S\) satisfying the following two axioms:
  1. For all \(a,b\in S\), exactly one holds: \(a=b\), \(a \lt b\), or \(a\gt b\).
  2. For all \(a,b,c\in S\), if \(a\lt b\) and \(b\lt c\) then \(a\lt c\).
An ordered field is a field \(F\) together with a total order \(\lt\) satisfying the following two axioms:
  1. For all \(a,b,c\in F\), if \(a\lt b\), then \(a+c\lt b+c\).
  2. For all \(a,b,c\in F\), if \(a\lt b\) and \(c \gt 0\), then \(a\cdot c\lt b\cdot c\).
Let \(F\) be an ordered field, and let \(a,b\in F\).
  1. \( 0 \lt 1\).
  2. If \(a \gt 1\), then \(a^2 \gt a\).
  3. If \(0\lt a \lt 1\), then \(a^2 \lt a\).
  4. If \(0\lt a\), then \(0 \lt 1/a\).
  5. If \(0\lt a \lt b\), then \(0 \lt b^{-1} \lt a^{-1}\).
Let \(F\) be an ordered field, and let \(a\in F\). The absolute value of \(a\), denoted by \(|a|\), is defined by \[ |a| = \begin{cases} \phantom{-}a, & \text{if }a \geq 0\\ -a, & \text{if } a \lt 0 \end{cases} \]
Let \(F\) be an ordered field, and let \(a,b,k\in F\) with \(k \gt 0\). Then
  1. \(-|a|\leq a \leq |a|\)
  2. \(|-a|=|a|\)
  3. \(|a|\lt k \iff -k \lt a \lt k\)
  4. \(|a| \gt k \iff a \gt k \text{ or }a \lt -k\)
  5. \(|ab| = |a|\cdot |b|\)
Let \(F\) be an ordered field, and let \(x,a,r\in F\) with \(r \gt 0\). Then \(|x -a| \lt r \iff x \in (a-r, a+r)\).
Let \(F\) be an ordered field, and let \(a,b\in F\). Then \(|a+b| \leq |a| + |b|\).
Let \(F\) be an ordered field, and let \(S\) be a nonempty subset of \(F\).
  1. An upper bound of \(S\) is an element \(M\in F\) such that \(M\geq s\) for all \(s\in S\). A lower bound of \(S\) is an element \(m\in F\) such that \(m\leq s\) for all \(s\in S\).
  2. \(S\) is bounded above if \(S\) has an upper bound. \(S\) is bounded below if \(S\) has a lower bound. \(S\) is bounded if it has both an upper and lower bound.
  3. A maximum of \(S\) is an upper bound of \(S\) which lies in \( S\). A minimum of \(S\) is a lower bound of \(S\) which lies in \( S\).
  4. A supremum of \(S\) is an element \(a\in F\) such that (1) \(a\) is an upper bound of \(S\) and (2) if \(b\) is an upper bound of \(S\) then \(a\leq b\). An infimum of \(S\) is an element \(c\in F\) such that (1) \(c\) is a lower bound of \(S\) and (2) if \(d\) is a lower bound of \(S\) then \(d\leq c\).
Let \(F\) be an ordered field. Then \(F\) is said to be complete if it satisfies the following Axiom of Completeness:
  1. Every subset of \(F\) which is bounded above has a supremum.

Note that a complete ordered field satisfies all eleven axioms F1 - F6, TO1 - TO2, OF1 - OF2, and AC.

There exists a unique complete ordered field.
The Real Numbers, denoted by \(\mathbb{R}\), are defined to be the unique complete ordered field whose existence is established by the above theorem.

The next four theorems are consequences of the Axiom of Completeness for \(\mathbb{R}\).

The set \(\mathbb{N}\) of natural numbers has no upper bound in \(\mathbb{R}\).
For any positive \(\epsilon\in \mathbb{R}\), no matter how small, there exists \(n\in\mathbb{N}\) such that \(0 \lt 1/n \lt \epsilon\).
For any two real numbers \(a, b\) with \(a \lt b\), there exists a rational number \(r\) such that \(a \lt r \lt b\).
For any two real numbers \(a, b\) with \(a \lt b\), there exists an irrational rational number \(r\) such that \(a \lt r \lt b\).
Cardinality
Sets \(A\) and \(B\) are said to have the same cardinality if there exists a bijection \(f:A \to B\). In this case we write \(A\sim B\).

Note that, in the above definition, if there exists a bijection \(f:A\to B\) , then there also exists a bijection \(g:B\to A\), namely, \(g=f^{-1}\). Thus the definition is symmetric in \(A\) and \(B\). (In fact it can be shown that \(\sim \) is an equivalence relation.)

A set \(A\) is
  1. finite if \(A=\emptyset\) or \(A \sim \{1,2,\ldots,n\}\) for some \(n\).
  2. infinite if \(A\) is not finite.
  3. countably infinite if \(A\sim \mathbb{N}\).
  4. countable if \(A\) is finite or countably infinite.
  5. uncountable if \(A\) is not countable.
\(\mathbb{N}\), \(\mathbb{Z}\), and \(\mathbb{Q}\) are countably infinite, but \(\mathbb{R}\) is uncountable.
Convergence of a Sequence

Informally, a sequence is an infinite list of real numbers \( (a_1, a_2, a_3,\ldots)\). Formally:

A sequence is a mapping (or function) from \(\mathbb{N}\) to \(\mathbb{R}\).

The mapping of this defintion is \(n \mapsto a_n \), where \(a_n\) is the \(n\)-th term of the sequence. We often denote the sequence by \((a_n)\).

Let \( L \in \mathbb{R}\). A neighborhood of \(L\), denoted by \( N_{\epsilon}(L)\), is a set of the form \((L-\epsilon,L+\epsilon) = \{x\in\mathbb{R} : |x - L| \lt \epsilon\} \) for some \(\epsilon>0\).
A sequence \( (a_n) \) converges to a real number \(L\) if, for every positive real number \(\epsilon\), there exists \(N \in \mathbb{N}\) such that \[ n\geq N \implies |a_n - L| \lt \epsilon \]

In other words, \( (a_n) \) converges to \(L\) if, for every \( \epsilon \gt 0\), however small, there is a point in the sequence beyond which all of the terms lie in the \(\epsilon\)-neighborhood \(N_{\epsilon}(L)\). We sometimes express convergence of \( (a_n) \) to \(L\) symbolically by writing \( (a_n) \to L\) or, alternatively, \(\lim\limits_{n\to \infty} a_n = L\).

If a sequence converges, then its limit is unique.
A sequence \( (a_1, a_2, a_3, \ldots) \) is said to be bounded if the set of terms \( \{ a_1, a_2, a_3, \ldots\} \) is bounded.
If a sequence converges, then it is bounded.
Let \( (a_n) \), \( (b_n) \), and \( (c_n) \) be sequences such that \(a_n \leq b_n \leq c_n \) for all \(n\in \mathbb{N}\). If \( (a_n) \to L\) and \( (c_n) \to L\), then \( (b_n) \to L\).
Supppose that \( (a_n) \) and \( (b_n) \) are convergent sequences and \(c\in \mathbb{R}\). Then
  1. \(\lim (c a_n) = c \lim (a_n)\).
  2. \(\lim (a_n + b_n) = \lim (a_n) + \lim (b_n)\).
  3. \(\lim (a_n \cdot b_n) = \lim (a_n) \cdot \lim (b_n)\).
  4. \(\lim (a_n / b_n) = \lim (a_n) / \lim (b_n)\), provided \(b_n \neq 0\) for all \(n\) and \( \lim (b_n) \neq 0\).
A sequence \( (a_n) \) of real numbers is
  1. increasing if \(a_n \leq a_{n+1}\) for all \(n\in \mathbb{N}\).
  2. decreasing if \(a_n \geq a_{n+1}\) for all \(n\in \mathbb{N}\).
  3. monotone if it is increasing or decreasing.
  4. strictly increasing if \(a_n \lt a_{n+1}\) for all \(n\in \mathbb{N}\).
  5. strictly decreasing if \(a_n \gt a_{n+1}\) for all \(n\in \mathbb{N}\).
If a sequence is monotone and bounded, then it converges.
Let \( (a_n) \) be a sequence and let \( n_1 \lt n_2 \lt n_3 \lt \cdots \) be an increasing sequence of natural numbers. Then \[ ( a_{n_1}, a_{n_2}, a_{n_3}, \ldots) \] is called a subsequence of \( (a_n) \) and is denoted by \( (a_{n_k}) \).
If a sequence \( (a_n) \) converges to \(L\), then any subsequence of \( (a_n) \) converges to \(L\) as well.
Topology of \( \mathbb{R} \)
A set \( U \) of real numbers is open if, for every \( a \in U\), there exists \( \epsilon > 0 \) for which \( N_{\epsilon}(a) \subseteq U \).
  1. The union of any collection of open sets is open.
  2. The intersection of any finite collection of open sets is open.
Let \(L \in \mathbb{R}\). A deleted neighborhood of \(L\), denoted by \( N^{*}_{\epsilon}(L)\), is a set of the form \( (L-\epsilon,L+\epsilon)\setminus \{ L \} = \{x\in\mathbb{R} : 0 \lt |x - L| \lt \epsilon\} \), for some \(\epsilon>0\).
Let \(A\) be a subset of \( \mathbb{R}\). A point \( x\in \mathbb{R} \) is called a limit point of \(A\) (or accumulation point of \(A\) ) if every deleted neighborhoood of \(x \) contains a point of \(A\).
Let \(A\) be a subset of \( \mathbb{R}\). A point \( x\in \mathbb{R} \) is a limit point of \(A\) if and only if \(x\) is the limit of a sequence of points in \(A \setminus \{ x\} \).
A set \(A \) of real numbers is closed if \(A^c\), the complement of \( A\) in \( \mathbb{R}\), is open.
  1. The union of any finite collection of closed sets is closed.
  2. The intersection of any collection of closed sets is closed.
A set is closed if and only if it contains all of its limit points.
Let \( A \) be a set of real numbers.
  1. The set of limit points of \( A\) is denoted by \( A' \).
  2. The closure of \( A\), denoted by \( \overline{A} \), is defined by \( \overline{A} = A \cup A' \).
A set \( A \) is closed if and only if \( A = \overline{A} \).
Let \( A \) be a set of real numbers.
  1. An open cover of \( A \) is a collection of open sets whose union contains \(A \).
  2. Given an open cover \( \mathcal{F} \) of \(A \), a finite subcover of \( \mathcal{F} \) is a finite subcollection of the open sets of \( \mathcal{F} \) whose union still contains \( A \).
A set \( A \) of real numbers is compact if every open cover of \( A \) has a finite subcover.
A set of real numbers is compact if and only if it is closed and bounded.
Topology of Metric Spaces
Let \( X \) be a nonempty set. A metric is function \( d : X\times X \to \mathbb{R} \) which satisfies the following four conditions: For all \( x, y, z \in X \),
  1. \( d(x,y) \geq 0 \).
  2. \(d(x,x) = 0 \), and \( d(x,y) = 0 \) only if \(x = y\).
  3. \( d(x,y) = d(y,x) \).
  4. \( d(x,y) \leq d(x,z) + d(z,y) \) (Triangle Inequality).
A metric space is a nonempty set \(X\) together with a metric \(d\). The metric space is denoted by \( (X,d)\).
Let \( (X,d ) \) be a metric space, let \(x\in X\), and let \( \epsilon>0 \). The \( \epsilon\)-neighborhood of \(x\), denoted by \(N_{\epsilon} (x) \), is \( \{y \in X : d(y,x) \lt \epsilon\} \).