User:Keith.U

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

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.

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

Exponential Function is Well-Defined/Real

Tail of Convergent Sequence

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

Union of Functions Theorem/Corollary

Union of Inverses of Mappings is Inverse of Union of Mappings

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

Book:Richard K. Miller/Ordinary Differential Equations

Uniqueness of Continuously Differentiable Solution to Initial Value Problem

Derivative of Exponential Function/Proof 4

Exponential of Zero/Proof 3

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

Natural Logarithm Function is Continuous/Proof 2

Defining Sequence of Natural Logarithm is Strictly Decreasing

Lower Bound of Natural Logarithm/Proof 3

Definition:Discontinuity of the First Kind

Discontinuity of Monotonic Function is Jump Discontinuity

Surjective Monotone Function is Continuous

Upper Bound of Natural Logarithm/Proof 2

Logarithm of 1 is 0/Proof 3

Sum of Logarithms/Natural Logarithm/Proof 3

Limit of Bounded Convergent Sequence is Bounded

Logarithm of Power/Natural Logarithm/Natural Power

Logarithm of Power/Natural Logarithm/Integer Power

Logarithm of Power/Natural Logarithm/Rational Power

Dini's Theorem

Monotone Real Function with Everywhere Dense Image is Continuous

Relative Difference between Infinite Set and Finite Set is Infinite

Set Difference over Subset

Subset of Empty Set iff Empty

Finite Subset of Metric Space is Closed

Monotone Real Function with Everywhere Dense Image is Continuous/Lemma

Defining Sequence of Natural Logarithm is Uniformly Convergent on Compact Sets

Multiplication of Positive Number by Real Number Greater than One

Definition:Real Interval/Bounded

Derivative of Uniformly Convergent Sequence of Differentiable Functions

Derivative of Natural Logarithm Function/Proof 4/Lemma

Derivative of Natural Logarithm Function/Proof 4

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

Sum of Geometric Progression/Corollary 2

Defining Sequence of Natural Logarithm is Strictly Decreasing

Lower Bound of Natural Logarithm/Proof 3

Definition:Jump Discontinuity

Discontinuity of Monotonic Function is Jump Discontinuity

Surjective Monotone Function is Continuous

Upper Bound of Natural Logarithm/Proof 2

Logarithm is Increasing

Logarithm of Power/Natural Logarithm/Rational Power

Logarithm of Power/Natural Logarithm/Integer Power

Logarithm of Power/Natural Logarithm/Natural Power

Limit of Bounded Convergent Sequence is Bounded

Sum of Logarithms/Natural Logarithm/Proof 3

Logarithm of 1 is 0/Proof 3

Upper Bound of Natural Logarithm/Proof 2

Subsets Inherit Uniform Convergence

Exponential with Base Greater than One is Strictly Increasing/Real Number

Defining Sequence of Natural Logarithm is Convergent

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

Definition:Pointwise Equicontinuous

Definition:Uniformly Equicontinuous

Uniformly Convergent iff Difference Under Supremum Norm Vanishes

Definition:Natural Logarithm/Positive Real/Definition 3

Exponential with Arbitrary Base is Continuous

Rational Sequence Increasing to Real Number

Rational Sequence Decreasing to Real Number

Definition:Less Than (Real Numbers)

Definition:Strictly Positive/Real Number/Definition 2

Exponent Combination Laws/Power of Power

Definition:Power (Algebra)

Inequality of Product of Unequal Numbers

Absolute Value Inequality is Compound Inequality/Corollary

Absolute Value Inequality is Compound Inequality

Exponent Combination Laws/Product of Powers

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

Number Equal to Zero iff Arbitrarily Small

Pasting Lemma/Corollary 2

Pasting Lemma/Corollary 1

Pasting Lemma

Euler's Number to Rational Power Permits Unique Continuous Extension

Exponential with Rational Power Permits Unique Continuous Extension

Exponential Tends to One as Power Tends to Zero/Rational Number

Exponential with Base between Zero and One Tends to One as Power Tends to Zero/Rational Number

Exponential with Base Greater Than One Tends to One as Power Tends to Zero/Rational Number

Exponential with Base Greater Than One Tends to One as Power Tends to Zero/Rational Number/Lemma

Exponential is Monotonic/Rational Number

Taking Additive Inverses Reverses Inequality

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

Inequality iff Difference is Positive

Exponential with Base Between Zero and One is Strictly Decreasing/Natural Number

Exponential with Base Greater than One is Strictly Increasing/Natural Number

Exponential with Base Between Zero and One is Strictly Decreasing/Integer

Exponential with Base Greater than One is Strictly Increasing/Integer

Exponential with Base Between Zero and One is Strictly Decreasing/Rational Number

Exponential with Base Greater than One is Strictly Increasing/Rational Number

Root of Reciprocal is Reciprocal of Root

Root is Strictly Increasing

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

Power is Well-Defined/Integer

Exponential Function is Continuous/Real Numbers/Proof 4

Definition:Exponential/Real

Moore-Osgood Theorem

Logarithm is Strictly Increasing and Strictly Concave/Corollary

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

Exponential Sequence is Eventually Increasing

Derivative of Exponential Function/Complex

Product of Uniformly Convergent Sequences of Bounded Functions is Uniformly Convergent

Uniformly Convergent Sequence of Bounded Functions is Uniformly Bounded

Derivative of Sequence of Holomorphic Functions

Definition:Bounded Sequence/Complex/Unbounded

Definition:Bounded Sequence/Complex

Uniformly Convergent Sequence Evaluated on Convergent Sequence

Exponential Sequence is Uniformly Convergent on Compact Sets

Identity Theorem

Vitali's Convergence Theorem

Montel's Theorem

Definition:Locally Bounded

Definition:Locally Bounded/Family of Mappings

Definition:Uniformly Bounded

Definition:Locally Bounded/Mapping