Proof of The Exponential Series Converges Absolutely, and Its Sum Is a Map on ℝ

Proof of (i). Let $(c_{k})=\big(1/(k-1)!\big)_{k\in\mathbb{N}}$ be the sequence of Power Series: the Sequence of Terms, the Coefficients 1/(k-1)!, and the Sum as a Map on the Set of Convergence (ii), and let $u\in\mathbb{R}$. For $k\in\mathbb{N}$, $u^{k-1}$ lies in $\mathbb{R}$ by Closure Facts in an Ordered Field: Negatives, Differences, Quotients, Absolute Values, Powers, Natural Numbers, Halves and Minima (iv), $\mathbb{R}$ being an ordered field by The Real Numbers Form an Ordered Field in Which Every Nonempty Set Bounded Above Has a Supremum (i); and $(k-1)!$ lies in $\mathbb{N}_{0}$, being the value at $k-1$ of the map of The Factorial as a Map on the Natural Numbers with Zero (i) (Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol (ii)), so its image read in $\mathbb{R}$ lies in $\mathbb{R}$ by Powers and Images of Natural Numbers in a Commutative Ring Lie in the Ring (ii), applied to the commutative ring $\mathbb{R}$ (The Structure of ℝ as a Field and an Ordered Ring (i)). Hence $c_{k}u^{k-1}=u^{k-1}/(k-1)!$, by Negatives, Differences, Reciprocals and Quotients (ii) and the commutative-ring laws of the field $\mathbb{R}$ (The Structure of ℝ as a Field and an Ordered Ring (i)). So a sequence in $\mathbb{R}$ has $k$-th term $u^{k-1}/(k-1)!$ for every $k\in\mathbb{N}$ exactly when it has $k$-th term $c_{k}u^{k-1}$ for every $k\in\mathbb{N}$, and by Power Series: the Sequence of Terms, the Coefficients 1/(k-1)!, and the Sum as a Map on the Set of Convergence (i) there is exactly one such sequence.

Proof of (ii). Let $u\in\mathbb{R}$. For $k\in\mathbb{N}$ we have $1+k\in\mathbb{N}$ by Natural Numbers Are the Successors in Omega: One Is Least and Not a Successor of a Natural Number, the Successor Is Injective, and N Is Closed under Addition and Multiplication (vi), so $1\le1+k$ by Arithmetic and Order of the Natural Numbers (x), and the difference $(1+k)-1$ is defined, with $(1+k)-1=k$ by The Difference of Two Natural Numbers with Zero (i), so the shifted series (of the shifted sequence, which exists by Sequences of Real Numbers Built from Given Ones: Partial Sums, Index Shifts, Predecessor Reindexing, Absolute Values and Termwise Operations (ii) with $p=1$) $\sum_{k=1}^{\infty}\frac{u^{(1+k)-1}}{((1+k)-1)!}$ is $\sum_{k=1}^{\infty}\frac{u^{k}}{k!}$, which converges absolutely by Standard Examples of Series: the Harmonic Series, the Reciprocals of the Squares, a Centred Quadratic Sum and the Exponential Series (iv); hence, by Elementary Properties of Series of Real Numbers: Linearity, Null Terms, the Cauchy Criterion, Index Shifts, Tails, Order and Telescoping (iv) applied with $p=1$ to the sequence of absolute values of the terms (Sequences of Real Numbers Built from Given Ones: Partial Sums, Index Shifts, Predecessor Reindexing, Absolute Values and Termwise Operations (iv)), $\sum_{k=1}^{\infty}\big|u^{k-1}/(k-1)!\big|$ converges. So $\sum_{k=1}^{\infty}\frac{u^{k-1}}{(k-1)!}$ converges absolutely, and it converges by Series of Real Numbers with Nonnegative Terms, Comparison, Absolute Convergence and the Geometric Series (iii). Its first term, for $k=1$, is $u^{0}/0!=1$: here $1-1=0$ by The Difference of Two Natural Numbers with Zero (i), as $1+0=1$ (Arithmetic of Addition on Omega: Recursion Rules, Associativity, Commutativity, Cancellation and Compatibility with the Order (i)); $u^{0}$ is the neutral element of the multiplication of $\mathbb{R}$ by Powers with Exponents in the Natural Numbers with Zero (ii), which is $1$ by A Binary Operation Has at Most One Neutral Element (i) and The Structure of ℝ as a Field and an Ordered Ring (i); $0!=1$ in $\mathbb{N}_{0}$ by The Factorial: Recursion, Positivity and Bounds by Powers (i), whose image in $\mathbb{R}$ is $1$ by The Image of the Natural Numbers with Zero in a Commutative Ring Respects Zero, One, Sums, Products, Differences, Powers, and Finite Sums and Products (i); and $1/1=1$ by Negatives, Differences, Reciprocals and Quotients (ii).

Proof of (iii). With $(c_{k})$ as in the proof of (i), for every $u\in\mathbb{R}$ the sequence $(c_{k}u^{k-1})_{k\in\mathbb{N}}$ of Power Series: the Sequence of Terms, the Coefficients 1/(k-1)!, and the Sum as a Map on the Set of Convergence (i) is the sequence of (i), so its series $\sum_{k=1}^{\infty}c_{k}u^{k-1}$ is $\sum_{k=1}^{\infty}\frac{u^{k-1}}{(k-1)!}$, which converges by (ii). Hence Power Series: the Sequence of Terms, the Coefficients 1/(k-1)!, and the Sum as a Map on the Set of Convergence (iii), applied with $E=\mathbb{R}$, gives exactly one map $h:\mathbb{R}\to\mathbb{R}$ with $h(u)=\sum_{k=1}^{\infty}\frac{u^{k-1}}{(k-1)!}$ for every $u\in\mathbb{R}$.