MATH201 - Real Analysis Fundamentals Flashcards

Question 1

Bolzano-Weierstrass Theorem

Accepted Answer

Every bounded sequence of real numbers has a convergent subsequence.

Question 2

Definition of a Sequence

Accepted Answer

A sequence of real numbers is a function $f: \mathbb{N} 	o \mathbb{R}$, where $f(n)$ is denoted by $a_n$. It is typically written as $(a_n)_{n=1}^\infty$ or $(a_n)$.

Question 3

Convergence of a Sequence

Accepted Answer

A sequence $(a_n)$ converges to a limit $L \in \mathbb{R}$ if for every $\varepsilon > 0$, there exists an $N \in \mathbb{N}$ such that for all $n > N$, $|a_n - L| < \varepsilon$. We write $\lim_{n 	o \infty} a_n = L$.

Question 4

Definition of a Series

Accepted Answer

A series is an expression of the form $\sum_{n=1}^\infty a_n = a_1 + a_2 + a_3 + \dots$. The $k$-th partial sum is $S_k = \sum_{n=1}^k a_n$.

Question 5

Convergence of a Series

Accepted Answer

A series $\sum_{n=1}^\infty a_n$ converges if its sequence of partial sums $(S_k)$ converges to a finite limit $S$. That is, $\lim_{k 	o \infty} S_k = S$.

Question 6

$\varepsilon-\delta$ Definition of Limit of a Function

Accepted Answer

A function $f: A o \mathbb{R}$ has a limit $L$ at $c$ (where $c$ is a limit point of $A$) if for every $\varepsilon > 0$, there exists a $\delta > 0$ such that for all $x \in A$ with $0 < |x - c| < \delta$, we have $|f(x) - L| < \varepsilon$. We write $\lim_{x o c} f(x) = L$.

Question 7

$\varepsilon-\delta$ Definition of Continuity at a Point

Accepted Answer

A function $f: A o \mathbb{R}$ is continuous at a point $c \in A$ if for every $\varepsilon > 0$, there exists a $\delta > 0$ such that for all $x \in A$ with $|x - c| < \delta$, we have $|f(x) - f(c)| < \varepsilon$. This is equivalent to $\lim_{x o c} f(x) = f(c)$.

Question 8

Definition of Uniform Continuity

Accepted Answer

A function $f: A o \mathbb{R}$ is uniformly continuous on $A$ if for every $\varepsilon > 0$, there exists a $\delta > 0$ such that for all $x, y \in A$ with $|x - y| < \delta$, we have $|f(x) - f(y)| < \varepsilon$. Note that $\delta$ depends only on $\varepsilon$, not on $x$ or $y$.

Question 9

Definition of the Derivative of a Function

Accepted Answer

The derivative of a function $f: (a,b) 	o \mathbb{R}$ at a point $c \in (a,b)$ is defined as $f'(c) = \lim_{h 	o 0} rac{f(c+h) - f(c)}{h}$, provided this limit exists.

Question 10

Definition of the Riemann Integral

Accepted Answer

For a bounded function $f:[a,b] 	o \mathbb{R}$, the Riemann integral $\int_a^b f(x) dx$ is defined as the common value of the supremum of the lower sums ($L(P,f)$) and the infimum of the upper sums ($U(P,f)$) over all partitions $P$ of $[a,b]$, provided these values are equal. If they are equal, $f$ is Riemann integrable.

Question 11

Heine-Borel Theorem

Accepted Answer

A subset $K$ of $\mathbb{R}$ is compact if and only if it is closed and bounded. (A set is compact if every open cover has a finite subcover).

Question 12

Intermediate Value Theorem (IVT)

Accepted Answer

If $f:[a,b] 	o \mathbb{R}$ is continuous and $y_0$ is any value between $f(a)$ and $f(b)$ (i.e., $f(a) < y_0 < f(b)$ or $f(b) < y_0 < f(a)$), then there exists some $c \in (a,b)$ such that $f(c) = y_0$.

Question 13

Extreme Value Theorem (EVT)

Accepted Answer

If $f:[a,b] 	o \mathbb{R}$ is continuous on a closed and bounded interval $[a,b]$, then $f$ attains both a maximum and a minimum value on $[a,b]$. That is, there exist $x_1, x_2 \in [a,b]$ such that $f(x_1) \le f(x) \le f(x_2)$ for all $x \in [a,b]$.

Question 14

Rolle's Theorem

Accepted Answer

If $f:[a,b] 	o \mathbb{R}$ is continuous on $[a,b]$ and differentiable on $(a,b)$, and if $f(a) = f(b)$, then there exists at least one $c \in (a,b)$ such that $f'(c) = 0$.

Question 15

Mean Value Theorem (MVT)

Accepted Answer

If $f:[a,b] 	o \mathbb{R}$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists at least one $c \in (a,b)$ such that $f'(c) = rac{f(b) - f(a)}{b - a}$.

Question 16

Fundamental Theorem of Calculus, Part 1 (FTC1)

Accepted Answer

If $f$ is continuous on $[a,b]$ and $F(x) = \int_a^x f(t) dt$ for $x \in [a,b]$, then $F$ is differentiable on $(a,b)$ and $F'(x) = f(x)$ for all $x \in (a,b)$.

Question 17

Fundamental Theorem of Calculus, Part 2 (FTC2)

Accepted Answer

If $f$ is Riemann integrable on $[a,b]$ and $F$ is any antiderivative of $f$ on $[a,b]$ (i.e., $F'(x) = f(x)$ for all $x \in (a,b)$), then $\int_a^b f(x) dx = F(b) - F(a)$.

Question 18

Definition of a Bounded Sequence

Accepted Answer

A sequence $(a_n)$ is bounded if there exists a real number $M > 0$ such that $|a_n| \le M$ for all $n \in \mathbb{N}$.

Question 19

Definition of a Monotonic Sequence

Accepted Answer

A sequence $(a_n)$ is monotonic if it is either increasing ($a_n \le a_{n+1}$ for all $n$) or decreasing ($a_n \ge a_{n+1}$ for all $n$).

Question 20

Monotone Convergence Theorem

Accepted Answer

Every bounded monotonic sequence of real numbers converges.

Question 21

Definition of a Cauchy Sequence

Accepted Answer

A sequence $(a_n)$ is a Cauchy sequence if for every $\varepsilon > 0$, there exists an $N \in \mathbb{N}$ such that for all $m, n > N$, $|a_m - a_n| < \varepsilon$.

Question 22

Completeness of $\mathbb{R}$

Accepted Answer

The real number system $\mathbb{R}$ is complete, meaning that every Cauchy sequence of real numbers converges to a real number. This is equivalent to the Monotone Convergence Theorem and the Bolzano-Weierstrass Theorem.

Question 23

Relationship between Differentiability and Continuity

Accepted Answer

If a function $f$ is differentiable at a point $c$, then $f$ is continuous at $c$. The converse is not true (e.g., $f(x) = |x|$ at $x=0$ is continuous but not differentiable).

MATH201 - Real Analysis Fundamentals

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