# Equivalence of Definitions of Euler's Number/Proof 2

## Theorem

The following definitions of the concept of Euler's Number are equivalent:

### Limit of Series

The series $\displaystyle \sum_{n \mathop = 0}^\infty \frac 1 {n!}$ converges to a limit.

This limit is Euler's number $e$.

### Limit of Sequence

The sequence $\sequence {x_n}$ defined as $x_n = \paren {1 + \dfrac 1 n}^n$ converges to a limit as $n$ increases without bound.

That limit is called Euler's Number and is denoted $e$.

### Base of Logarithm

The number $e$ can be defined as the number satisfied by:

$\ln e = 1$.

where $\ln e$ denotes the natural logarithm of $e$.

That $e$ is unique follows from Logarithm is Strictly Increasing.

### Exponential Function

The number $e$ can be defined as the number satisfied by:

$e := \exp 1 = e^1$

where $\exp 1$ denotes the exponential function of $1$.

## Proof

### 1 implies 2

$\Box$

### 2 implies 3

$\Box$

### 3 implies 4

Let $e$ be the unique solution to the equation $\ln \left({x}\right) = 1$.

We want to show that $\exp \left({1}\right) = e$, where $\exp$ is the exponential function.

 $\displaystyle \exp 1 = 0$ $\iff$ $\displaystyle \ln \left({ \exp 1 }\right) = \ln \left({ e }\right)$ $\quad$ Logarithm is Injective $\quad$ $\displaystyle$ $\iff$ $\displaystyle 1 = \ln \left({ e }\right)$ $\quad$ Exponential is Inverse of Logarithm and Inverse of Inverse $\quad$

where the final equation holds by hypothesis.

Hence the result.

$\Box$

### 4 implies 1

Let $e = \exp 1$, where $\exp$ denotes the exponential function.

We want to show that:

$\displaystyle \sum_{n \mathop = 0}^\infty \frac 1 {n!} = e$
$\displaystyle \sum_{n \mathop = 0}^\infty \frac 1 {n!} = \exp 1$

And $\exp 1 = e$ by hypothesis.

Hence the result.

$\blacksquare$