Equivalence of Definitions of Euler's Number/Proof 2

1 implies 2
See Euler's Number: Limit of Sequence implies Limit of Series.

2 implies 3
See Euler's Number: Limit of Sequence implies Base of Logarithm.

3 implies 4
Let $e$ be the unique solution to the equation $\ln x = 1$.

We want to show that $\exp 1 = e$, where $\exp$ is the exponential function.

where the final equation holds by hypothesis.

Hence the result.

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.