# User:Keith.U

Undergraduate mathematics\statistics student. Areas of study: Real analysis, metric spaces, probability theory.

## Contents

## My Current Focus

My current focus on proofwiki is developing a well-rounded and rigorous exposition of the exponential and (necessarily) logarithmic functions. Some of the approaches are rather tedious, and thus I'm forced to make my own proofs for much of the minutia, a recipe for certain disaster. If I've made a mistake somewhere, or you think I could be approaching a certain subject with more ease, please let me know!

## Pages I've Created or Refactored

This exists as a quick reference for me to check over previous pages I've created or done major edits on, so that I may keep track of what I've done and round out entries made while perhaps overtired. If a page you have created or done significant work on appears here, please don't take this to mean I'm trying to claim any credit for your work. Again, this is so I may easily check for any "proofread", "tidy" or "refactor" tags that might arise, so I may correct my mistakes. A star indicates that it is up to house standards.

### Books*

Book:John B. Conway/Functions of One Complex Variable/Second Edition

Book:Eberhard Freitag/Complex Analysis

Book:E.C. Titchmarsh/The Theory of Functions/Second Edition

Book:Richard K. Miller/Ordinary Differential Equations

Book:E.C. Titchmarsh/The Theory of Functions/Second Edition

Book:Charles C. Pugh/Real Mathematical Analysis/Second Edition

Book:Robert G. Bartle/Introduction to Real Analysis/Fourth Edition

Book:Brian S. Thomson/Symmetric Properties of Real Functions

### Definitions*

Definition:Norm of Subdivision

Definition:Discontinuity of the First Kind

Definition:Real Interval/Bounded

Definition:Pointwise Equicontinuous

Definition:Uniformly Equicontinuous

Definition:Power (Algebra)#Real Numbers

Definition:Natural Logarithm/Positive Real/Definition 3

Definition:Less Than (Real Numbers)

Definition:Strictly Positive/Real Number/Definition 2

Definition:Real Exponential Function

Definition:Bounded Sequence/Complex/Unbounded

Definition:Bounded Sequence/Complex

Definition:Locally Bounded/Family of Mappings

Definition:Locally Bounded/Mapping

Definition:Euler's Number/Base of Exponential

Definition:Strictly Midpoint-Convex

Definition:Strictly Midpoint-Concave

### Theorems

#### Set Theory*

Relative Difference between Infinite Set and Finite Set is Infinite

#### Mapping Theory*

Union of Functions Theorem/Corollary

Union of Inverses of Mappings is Inverse of Union of Mappings

#### Real Numbers/Order

Multiplication of Positive Number by Real Number Greater than One*

Power Function on Base Greater than One is Strictly Increasing/Real Number*

Rational Sequence Increasing to Real Number*

Rational Sequence Decreasing to Real Number*

Inequality of Product of Unequal Numbers*

Number Equal to Zero iff Arbitrarily Small*

Inequality iff Difference is Positive

Exponential is Monotonic/Rational Number*

Power of Positive Real Number is Positive/Natural Number

Power of Positive Real Number is Positive/Integer*

Power of Positive Real Number is Positive/Rational Number*

Power Function on Base between Zero and One is Strictly Decreasing/Natural Number*

Power Function on Base Greater than One is Strictly Increasing/Natural Number*

Power Function on Base between Zero and One is Strictly Decreasing/Integer*

Power Function on Base Greater than One is Strictly Increasing/Integer*

Power Function on Base between Zero and One is Strictly Decreasing/Rational Number*

Power Function on Base Greater than One is Strictly Increasing/Rational Number*

Root of Reciprocal is Reciprocal of Root*

Power Function is Strictly Increasing over Positive Reals/Natural Exponent*

Real Number between Zero and One is Greater than Power/Natural Number*

Power is Well-Defined/Rational

#### Transcendental Functions

** Exponential Functions **

Equivalence of Definitions of Real Exponential Function/Proof 2*

Exponential of Real Number is Strictly Positive/Proof 5/Lemma*

Exponential of Real Number is Strictly Positive/Proof 5*

Exponential of Real Number is Strictly Positive/Proof 4*

Exponential of Real Number is Strictly Positive/Proof 3*

Exponential of Real Number is Strictly Positive/Proof 2*

Exponential of Real Number is Strictly Positive/Proof 1*

Exponential of Product/Proof 3*

Exponential of Sum/Real Numbers/Proof 5*

Exponential of Sum/Real Numbers/Proof 4*

Power Function on Strictly Positive Base is Convex

Derivative of Exponential Function/Proof 5/Lemma*

Derivative of Exponential Function/Proof 5

Exponential Function is Well-Defined/Real*

Exponential Function is Well-Defined/Real/Proof 1*

Exponential Function is Well-Defined/Real/Proof 2

Exponential Function is Well-Defined/Real/Proof 3*

Exponential Function is Well-Defined/Real/Proof 4*

Exponential Function is Well-Defined/Real/Proof 5*

Exponent Combination Laws/Product of Powers/Proof 2/Lemma*

Derivative of Exponential Function/Proof 4*

Exponential Function is Continuous/Real Numbers/Proof 5*

Exponential Function is Well-Defined/Real/Proof 5*

Exponential Function is Well-Defined/Real/Proof 4/Lemma*

Exponent Combination Laws/Power of Power*

Exponent Combination Laws/Product of Powers*

Exponent Combination Laws/Product of Powers/Proof 2/Lemma*

Euler's Number to Rational Power permits Unique Continuous Extension*

Power Function to Rational Power permits Unique Continuous Extension

Power Function tends to One as Power tends to Zero/Rational Number

Power Function on Base between Zero and One Tends to One as Power Tends to Zero/Rational Number

Power Function on Base greater than One tends to One as Power tends to Zero/Rational Number*

Power Function on Base greater than One tends to One as Power tends to Zero/Rational Number/Lemma

Exponential Function is Continuous/Real Numbers/Proof 4*

Euler's Number: Limit of Sequence implies Base of Logarithm*

Exponential Sequence is Eventually Increasing*

Exponential Sequence is Eventually Strictly Positive

Derivative of Exponential Function/Complex*

Exponential Sequence is Uniformly Convergent on Compact Sets

** Natural Logarithm **

Natural Logarithm Function is Continuous/Proof 2

Defining Sequence of Natural Logarithm is Strictly Decreasing

Lower Bound of Natural Logarithm/Proof 3

Upper Bound of Natural Logarithm/Proof 2

Natural Logarithm of 1 is 0/Proof 3

Sum of Logarithms/Natural Logarithm/Proof 3

Logarithm of Power/Natural Logarithm/Natural Power

Logarithm of Power/Natural Logarithm/Integer Power

Logarithm of Power/Natural Logarithm/Rational Power

Defining Sequence of Natural Logarithm is Uniformly Convergent on Compact Sets

Derivative of Natural Logarithm Function/Proof 4/Lemma

Derivative of Natural Logarithm Function/Proof 4

Defining Sequence of Natural Logarithm is Strictly Decreasing

Lower Bound of Natural Logarithm/Proof 3

Upper Bound of Natural Logarithm/Proof 2

Logarithm of Power/Natural Logarithm/Rational Power

Logarithm of Power/Natural Logarithm/Integer Power

Logarithm of Power/Natural Logarithm/Natural Power

Sum of Logarithms/Natural Logarithm/Proof 3

Natural Logarithm of 1 is 0/Proof 3

Upper Bound of Natural Logarithm/Proof 2

Defining Sequence of Natural Logarithm is Convergent

Logarithm is Strictly Increasing and Strictly Concave/Corollary

Natural Logarithm as Derivative of Exponential at Zero

#### Real Analysis

Ordering of Series of Ordered Sequences

Product of Increasing Positive Functions is Increasing

Uniqueness of Continuously Differentiable Solution to Initial Value Problem

Discontinuity of Monotonic Function is Jump Discontinuity

Surjective Monotone Function is Continuous

Limit of Bounded Convergent Sequence is Bounded

Monotone Real Function with Everywhere Dense Image is Continuous

Finite Subset of Metric Space is Closed

Monotone Real Function with Everywhere Dense Image is Continuous/Lemma

Derivative of Uniformly Convergent Sequence of Differentiable Functions

Sum of Geometric Progression/Corollary 2

Discontinuity of Monotonic Function is Jump Discontinuity

Surjective Monotone Function is Continuous

Limit of Bounded Convergent Sequence is Bounded

Subsets Inherit Uniform Convergence

Uniformly Convergent iff Difference Under Supremum Norm Vanishes

Product of Uniformly Convergent Sequences of Bounded Functions is Uniformly Convergent

Uniformly Convergent Sequence of Bounded Functions is Uniformly Bounded

Uniformly Convergent Sequence Evaluated on Convergent Sequence

Continuous Midpoint-Convex Function is Convex

Continuous Strictly Midpoint-Convex Function is Strictly Convex

Continuous Midpoint-Concave Function is Concave

Continuous Strictly Midpoint-Concave Function is Strictly Concave

Power Function on Base Greater than One is Strictly Convex