Equivalence of Definitions of Euler's Number/Proof 2

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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 $\left \langle {x_n} \right \rangle$ defined as $x_n = \left({1 + \dfrac 1 n}\right)^n$ 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

See Euler's Number: Limit of Sequence Equals Limit of Series.

$\Box$

2 implies 3

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

$\Box$

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:

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

By definition of $\exp$:

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

And $\exp 1 = e$ by hypothesis.

Hence the result.

$\blacksquare$