MATH202 - Real Analysis: Continuity, Limits & Metric Spaces Flashcards

Question 1

Cauchy Sequence

Accepted Answer

A sequence $(x_n)$ in a metric space $(M, d)$ is a **Cauchy sequence** if terms become arbitrarily close to each other:
$$\forall \epsilon > 0, \exists N \in \mathbb{N} 	ext{ such that } \forall n, m \ge N, d(x_n, x_m) < \epsilon$$

Question 2

Definition of a Metric Space

Accepted Answer

A metric space is a pair $(M, d)$ where $M$ is a set and $d: M 	imes M 	o \mathbb{R}_{\ge 0}$ satisfies for all $x, y, z \in M$:
1. **Identity of Indiscernibles**: $d(x, y) = 0 \iff x = y$
2. **Symmetry**: $d(x, y) = d(y, x)$
3. **Triangle Inequality**: $d(x, z) \le d(x, y) + d(y, z)$

Question 3

Open Set in a Metric Space

Accepted Answer

A subset $U \subseteq M$ in metric space $(M, d)$ is **open** if for every $x \in U$, there exists an $r > 0$ such that the open ball $B_r(x) = \{y \in M : d(x, y) < r\} \subseteq U$.

Question 4

Closed Set in a Metric Space

Accepted Answer

A subset $F \subseteq M$ is **closed** if its complement $M \setminus F$ is open. Equivalently, $F$ is closed if it contains all of its limit points: whenever $x_n \in F$ and $x_n 	o x$, then $x \in F$.

Question 5

Epsilon-Delta Definition of a Limit

Accepted Answer

A function $f: D o \mathbb{R}$ has limit $L$ at an accumulation point $c$ of $D$, written $\lim_{x o c} f(x) = L$, if: $$\forall \epsilon > 0, \exists \delta > 0 ext{ such that } 0 < |x - c| < \delta \implies |f(x) - L| < \epsilon$$

Question 6

Sequential Criterion for Limits

Accepted Answer

$\lim_{x 	o c} f(x) = L$ if and only if for every sequence $(x_n)$ in $D \setminus \{c\}$ converging to $c$, the image sequence $(f(x_n))$ converges to $L$.

Question 7

Definition of Continuity at a Point

Accepted Answer

A function $f: D o \mathbb{R}$ is continuous at $c \in D$ if: $$\forall \epsilon > 0, \exists \delta > 0 ext{ such that } \forall x \in D, |x - c| < \delta \implies |f(x) - f(c)| < \epsilon$$ Equivalently, $f$ is continuous at $c$ if $\lim_{x o c} f(x) = f(c)$.

Question 8

Uniform Continuity

Accepted Answer

A function $f: D o \mathbb{R}$ is **uniformly continuous** on $D$ if: $$\forall \epsilon > 0, \exists \delta > 0 ext{ such that } \forall x, y \in D, |x - y| < \delta \implies |f(x) - f(y)| < \epsilon$$ Unlike pointwise continuity, $\delta$ depends *only* on $\epsilon$ and not on the choice of point.

Question 9

Complete Metric Space

Accepted Answer

A metric space $(M, d)$ is **complete** if every Cauchy sequence in $M$ converges to a point inside $M$.

*Example*: $\mathbb{R}$ with Euclidean metric is complete; $\mathbb{Q}$ is incomplete.

Question 10

Bolzano-Weierstrass Theorem

Accepted Answer

Every bounded sequence in $\mathbb{R}^n$ contains a convergent subsequence.

Question 11

Heine-Borel Theorem

Accepted Answer

In $\mathbb{R}^n$ with the Euclidean metric, a subset $K \subseteq \mathbb{R}^n$ is **compact** if and only if $K$ is **closed and bounded**.

Question 12

Compactness via Open Covers

Accepted Answer

A subset $K \subseteq M$ is **compact** if every open cover of $K$ has a finite subcover. That is, if $K \subseteq \bigcup_{\alpha \in A} U_\alpha$ with open $U_\alpha$, there exist $\alpha_1, \dots, \alpha_k \in A$ such that $K \subseteq \bigcup_{i=1}^k U_{\alpha_i}$.

Question 13

Sequential Compactness

Accepted Answer

A metric space $M$ is **sequentially compact** if every sequence in $M$ has a subsequence that converges to a limit in $M$. In general metric spaces, sequential compactness is equivalent to open cover compactness.

Question 14

Extreme Value Theorem

Accepted Answer

If $K$ is a compact metric space and $f: K 	o \mathbb{R}$ is continuous, then $f$ is bounded and attains its absolute maximum and minimum on $K$.

Question 15

Heine-Cantor Theorem

Accepted Answer

If $K$ is a compact metric space and $f: K 	o M'$ is continuous, then $f$ is **uniformly continuous** on $K$.

Question 16

Connected Metric Space

Accepted Answer

A metric space $M$ is **connected** if it cannot be partitioned into two non-empty, disjoint open sets. Equivalently, the only subsets of $M$ that are both open and closed are $\emptyset$ and $M$.

Question 17

Intermediate Value Theorem (IVT)

Accepted Answer

If $f: [a, b] 	o \mathbb{R}$ is continuous and $y$ is any value strictly between $f(a)$ and $f(b)$, then there exists at least one $c \in (a, b)$ such that $f(c) = y$.

Question 18

Contraction Mapping

Accepted Answer

A map $T: M 	o M$ on metric space $(M, d)$ is a **contraction** if there exists a constant $k \in [0, 1)$ such that:
$$d(T(x), T(y)) \le k \cdot d(x, y) \quad \forall x, y \in M$$

Question 19

Banach Fixed-Point Theorem

Accepted Answer

Let $(M, d)$ be a non-empty **complete metric space** and let $T: M 	o M$ be a contraction mapping. Then $T$ has a unique fixed point $x^* \in M$ satisfying $T(x^*) = x^*$.

Question 20

Supremum Property (Completeness Axiom of R)

Accepted Answer

Every non-empty subset $S \subseteq \mathbb{R}$ that is bounded above has a least upper bound (supremum) in $\mathbb{R}$, denoted $\sup S$.

MATH202 - Real Analysis: Continuity, Limits & Metric Spaces

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