Intro Analysis - Introduction to Analysis

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Question

Determine whether the following statement is true or false:

Let , , , and $d\ \epsilon\ \mathbb{R}$ . If and then .

Answer

Determine this statement is false by showing a contradiction when actual values are used.

Let

First make sure the inequalities hold true.

$\a<b \1<2$

and

$\c<d<0 \-3<-2<0$

Now find the products.

$\ac<bd \(1)(-3)<(2)(-2) \-3 less -4$

Therefore, the statement is false.

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Question

Determine whether the following statement is true or false:

If is a nonempty subset of $\mathbb{N}$ , then has a finite infimum and it is an element of .

Answer

According to the Well-Ordered Principal this statement is true. The following proof illuminate its truth.

Suppose $A\subseteq \mathbb{N}$ is nonempty. From there, it is known that is bounded above, by .

Therefore, by the Completeness Axiom the supremum of exists.

Furthermore, if $A\subset \mathbb{Z}$ has a supremum, then $\text{sup}A\ \epsilon\ A$ , thus in this particular case $\text{sup}(-A)\ \epsilon\ -A$ .

Thus by the Reflection Principal,

$\text{inf}A=-\text{sup}(-A)$

exists and

$\text{inf}A\ \epsilon\ -(-A)=A$ .

Therefore proving the statement in question true.

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Question

What term does the following define.

A sequence of sets $\begin{Bmatrix} I_n \end{Bmatrix}_{n\ \epsilon\ \mathbb{N}}$ is __________ if and only if $I_1\supseteq I_2\supseteq ...$ .

Answer

This statement:

A sequence of sets $\begin{Bmatrix} I_n \end{Bmatrix}_{n\ \epsilon\ \mathbb{N}}$ is __________ if and only if $I_1\supseteq I_2\supseteq ...$

is the definition of nested.

This means that the sequence for all elements, for which belongs to the natural numbers, is considered a nested set if and only if the subsequent sets are subsets of it.

Other theorems in intro analysis build off this understanding.

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Question

Identify the following property.

On the space $\mathbb{R}\times \mathbb{R}$ where , $b\ \epsilon\ \mathbb{R}$ only one of the following statements holds true , , or .

Answer

The real number system, $\mathbb{R}\times \mathbb{R}$ contains order axioms that show relations and properties of the system that add completeness to the ordered field algebraic laws.

The properties are as follows.

Trichotomy Property:

Given , $b\ \epsilon\ \mathbb{R}$ only one of the following statements holds true , , or .

Transitive Property:

For , , and $c\ \epsilon\ \mathbb{R}$ where and then this implies .

Additive Property:

For , , and $c\ \epsilon\ \mathbb{R}$ where and $c\ \epsilon\ \mathbb{R}$ then this implies .

Multiplicative Properties:

For , , and $c\ \epsilon\ \mathbb{R}$ where and then this implies and and then this implies .

Therefore looking at the options the Trichotomy Property identifies the property in this particular question.

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Question

Identify the following property.

For , , and $c\ \epsilon\ \mathbb{R}$ where and then this implies .

Answer

The real number system, $\mathbb{R}\times \mathbb{R}$ contains order axioms that show relations and properties of the system that add completeness to the ordered field algebraic laws.

The properties are as follows.

Trichotomy Property:

Given , $b\ \epsilon\ \mathbb{R}$ only one of the following statements holds true , , or .

Transitive Property:

For , , and $c\ \epsilon\ \mathbb{R}$ where and then this implies .

Additive Property:

For , , and $c\ \epsilon\ \mathbb{R}$ where and $c\ \epsilon\ \mathbb{R}$ then this implies .

Multiplicative Properties:

For , , and $c\ \epsilon\ \mathbb{R}$ where and then this implies and and then this implies .

Therefore looking at the options the Transitive Property identifies the property in this particular question.

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Question

Identify the following property.

For , , and $c\ \epsilon\ \mathbb{R}$ where and $c\ \epsilon\ \mathbb{R}$ then this implies .

Answer

The real number system, $\mathbb{R}\times \mathbb{R}$ contains order axioms that show relations and properties of the system that add completeness to the ordered field algebraic laws.

The properties are as follows.

Trichotomy Property:

Given , $b\ \epsilon\ \mathbb{R}$ only one of the following statements holds true , , or .

Transitive Property:

For , , and $c\ \epsilon\ \mathbb{R}$ where and then this implies .

Additive Property:

For , , and $c\ \epsilon\ \mathbb{R}$ where and $c\ \epsilon\ \mathbb{R}$ then this implies .

Multiplicative Properties:

For , , and $c\ \epsilon\ \mathbb{R}$ where and then this implies and and then this implies .

Therefore looking at the options the Additive Property identifies the property in this particular question.

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Question

Identify the following property.

For , , and $c\ \epsilon\ \mathbb{R}$ where and then this implies and and then this implies .

Answer

The real number system, $\mathbb{R}\times \mathbb{R}$ contains order axioms that show relations and properties of the system that add completeness to the ordered field algebraic laws.

The properties are as follows.

Trichotomy Property:

Given , $b\ \epsilon\ \mathbb{R}$ only one of the following statements holds true , , or .

Transitive Property:

For , , and $c\ \epsilon\ \mathbb{R}$ where and then this implies .

Additive Property:

For , , and $c\ \epsilon\ \mathbb{R}$ where and $c\ \epsilon\ \mathbb{R}$ then this implies .

Multiplicative Properties:

For , , and $c\ \epsilon\ \mathbb{R}$ where and then this implies and and then this implies .

Therefore looking at the options the Multiplicative Properties identifies the property in this particular question.

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Question

Determine whether the following statement is true or false:

Some power series such as have a possible radius of convergence that can be found by computing the roots of the coefficients of .

Answer

This statement is false because every power series has a radius of convergence that is found by computing the roots the series coefficients.

The proof of one case is as follows:

When $x\ \epsilon\ \mathbb{R}$ is fixed, , and $\rho:=\frac{1}{\lim \text{sup}_{k\rightarrow \infty}|a_k|^{\frac{1}{k}}}$

with the assumption that $\frac{1}{\infty}=0$ and $\frac{1}{0}=\infty$ the Root Test is applied to the series .

When $\rho =0$ by the assumption and the notation that

$r(x):=\lim \sup_{k\rightarrow \infty}|a_k(x-x_0)^k|^{\frac{1}{k}}=|x-x_0\cdot \lim \sup_{k\rightarrow \infty}|a_k|^\frac{1}{k}$

implies that $r(x)=\infty>1$ therefore by the Root Test, does not converge for any and thus the radius of convergence of is $R=0=\rho$ .

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Question

What conditions are necessary to prove that the upper and lower integrals of a bounded function exist?

Answer

Using the definition for Riemann sums to define the upper and lower integrals of a function answers the question.

According the the Riemann sum where represents the upper integral and the following are defined:

1. The upper integral of on is

$(U)\int_a^bf(x)dx:=\text{inf}\begin{Bmatrix} U(f,p)) \end{Bmatrix}$ where is a partition of .

2. The lower integral of on is

$(L)\int_a^bf(x)dx:=\text{sup}\begin{Bmatrix} L(f,p)) \end{Bmatrix}$ where is a partition of .

3. If 1 and 2 are the same then the integral is said to be

$\int_a^bf(x):=(U)\int_a^bf(x)dx=(L)\int_a^bf(x)dx$

if and only if , $b\ \epsilon\ \mathbb{R}$ , , and $f:[a,b]\rightarrow \mathbb{R}$ be bounded.

Therefore the necessary condition for the proving the upper and lower integrals of a bounded function exists is if and only if , $b\ \epsilon\ \mathbb{R}$ , , and $f:[a,b]\rightarrow \mathbb{R}$ be bounded.

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Question

What term has the following definition.

, $b\ \epsilon\ \mathbb{R}$ and . Over the interval is a set of points $P=\begin{Bmatrix} x_0,x_1,...,x_n \end{Bmatrix}$ such that

Answer

By definition

If , $b\ \epsilon\ \mathbb{R}$ and .

A partition over the interval is a set of points $P=\begin{Bmatrix} x_0,x_1,...,x_n \end{Bmatrix}$ such that

.

Therefore, the term that describes this statement is partition.

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Question

What term has the following definition.

The __________ of a partition $P=\begin{Bmatrix} x_0,x_1,...,x_n \end{Bmatrix}$ is

$||P||=\max_{1\leq j\leq n}|x_j-x_{j-1}|$

Answer

By definition

If , $b\ \epsilon\ \mathbb{R}$ and .

A partition over the interval is a set of points $P=\begin{Bmatrix} x_0,x_1,...,x_n \end{Bmatrix}$ such that

.

Furthermore,

The norm of the partition

$P=\begin{Bmatrix} x_0,x_1,...,x_n \end{Bmatrix}$ is

$||P||=\max_{1\leq j\leq n}|x_j-x_{j-1}|$

Therefore, the term that describes this statement is norm.

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