Equivalence of Definitions of Real Exponential Function

Theorem
The following definitions of the exponential function are equivalent.

1 implies 5
This proves that $y$ is a solution.

It remains to be proven that $y$ fulfils the initial condition:

5 implies 1
To solve for $C$, put $\left({x_0, y_0}\right) = \left({0, 1}\right)$:

3 implies 4
From the Binomial Theorem:

From Power of Number less than One, this converges to:


 * $\exp x - \frac {x^0} {0!} + \frac {x^1} {1!} + \frac {x^2} {2!} + \frac {x^3} {3!} + \cdots = 0$

as $n \to +\infty$:

Compare Series of Power over Factorial Converges.

3 implies 5
The application of Derivative of Exponential at Zero is not circular as the referenced proof does not depend on $D_x \exp x = \exp x$.

Also see

 * Equivalence of Definitions of Euler's Number
 * Equivalence of Definitions of Natural Logarithm